2015

• Paper 1, Section I, $3 \mathbf{F}$

  #Analysis I

  2015

  Find the following limits: (a) $\lim _{x \rightarrow 0} \frac{\sin x}{x}$ (b) $\lim _{x \rightarrow 0}(1+x)^{1 / x}$ (c) $\lim _{x \rightarrow \infty} \frac{(1+x)^{\frac{x}{1+x}} \cos ^{4} x}{e^{x}}$

  Carefully justify your answers.

  [You may use standard results provided that they are clearly stated.]

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• Paper 1, Section I, E

  #Analysis I

  2015

  Let $\sum_{n \geqslant 0} a_{n} z^{n}$ be a complex power series. State carefully what it means for the power series to have radius of convergence $R$, with $0 \leqslant R \leqslant \infty$.

  Find the radius of convergence of $\sum_{n \geqslant 0} p(n) z^{n}$, where $p(n)$ is a fixed polynomial in $n$ with coefficients in $\mathbb{C}$.

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• Paper 1, Section II, F

  #Analysis I

  2015

  Let $\left(a_{n}\right),\left(b_{n}\right)$ be sequences of real numbers. Let $S_{n}=\sum_{j=1}^{n} a_{j}$ and set $S_{0}=0$. Show that for any $1 \leqslant m \leqslant n$ we have

  $$\sum_{j=m}^{n} a_{j} b_{j}=S_{n} b_{n}-S_{m-1} b_{m}+\sum_{j=m}^{n-1} S_{j}\left(b_{j}-b_{j+1}\right)$$

  Suppose that the series $\sum_{n \geqslant 1} a_{n}$ converges and that $\left(b_{n}\right)$ is bounded and monotonic. Does $\sum_{n \geqslant 1} a_{n} b_{n}$ converge?

  Assume again that $\sum_{n \geqslant 1} a_{n}$ converges. Does $\sum_{n \geqslant 1} n^{1 / n} a_{n}$ converge?

  Justify your answers.

  [You may use the fact that a sequence of real numbers converges if and only if it is a Cauchy sequence.]

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• Paper 1, Section II, 10D

  #Analysis I

  2015

  (a) For real numbers $a, b$ such that $a<b$, let $f:[a, b] \rightarrow \mathbb{R}$ be a continuous function. Prove that $f$ is bounded on $[a, b]$, and that $f$ attains its supremum and infimum on $[a, b]$.

  (b) For $x \in \mathbb{R}$, define

  $$g(x)=\left\{\begin{array}{ll} |x|^{\frac{1}{2}} \sin (1 / \sin x), & x \neq n \pi \\ 0, & x=n \pi \end{array} \quad(n \in \mathbb{Z})\right.$$

  Find the set of points $x \in \mathbb{R}$ at which $g(x)$ is continuous.

  Does $g$ attain its supremum on $[0, \pi] ?$

  Does $g$ attain its supremum on $[\pi, 3 \pi / 2]$ ?

  Justify your answers.

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• Paper 1, Section II, $11 D$

  #Analysis I

  2015

  (i) State and prove the intermediate value theorem.

  (ii) Let $f:[0,1] \rightarrow \mathbb{R}$ be a continuous function. The chord joining the points $(\alpha, f(\alpha))$ and $(\beta, f(\beta))$ of the curve $y=f(x)$ is said to be horizontal if $f(\alpha)=f(\beta)$. Suppose that the chord joining the points $(0, f(0))$ and $(1, f(1))$ is horizontal. By considering the function $g$ defined on $\left[0, \frac{1}{2}\right]$ by

  $$g(x)=f\left(x+\frac{1}{2}\right)-f(x)$$

  or otherwise, show that the curve $y=f(x)$ has a horizontal chord of length $\frac{1}{2}$ in $[0,1]$. Show, more generally, that it has a horizontal chord of length $\frac{1}{n}$ for each positive integer $n$.

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• Paper 1, Section II, E

  #Analysis I

  2015

  Let $f:[0,1] \rightarrow \mathbb{R}$ be a bounded function, and let $\mathcal{D}_{n}$ denote the dissection $0<\frac{1}{n}<\frac{2}{n}<\cdots<\frac{n-1}{n}<1$ of $[0,1]$. Prove that $f$ is Riemann integrable if and only if the difference between the upper and lower sums of $f$ with respect to the dissection $\mathcal{D}_{n}$ tends to zero as $n$ tends to infinity.

  Suppose that $f$ is Riemann integrable and $g: \mathbb{R} \rightarrow \mathbb{R}$ is continuously differentiable. Prove that $g \circ f$ is Riemann integrable.

  [You may use the mean value theorem provided that it is clearly stated.]

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• Paper 2, Section I, B

  #Differential Equations

  2015

  Find the general solution of the equation

  $$\frac{d y}{d x}-2 y=e^{\lambda x}$$

  where $\lambda$ is a constant not equal to 2 .

  By subtracting from the particular integral an appropriate multiple of the complementary function, obtain the limit as $\lambda \rightarrow 2$ of the general solution of $(*)$ and confirm that it yields the general solution for $\lambda=2$.

  Solve equation $(*)$ with $\lambda=2$ and $y(1)=2$.

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• Paper 2, Section $I$, B

  #Differential Equations

  2015

  Find the general solution of the equation

  $$2 \frac{d y}{d t}=y-y^{3} .$$

  Compute all possible limiting values of $y$ as $t \rightarrow \infty$.

  Find a non-zero value of $y(0)$ such that $y(t)=y(0)$ for all $t$.

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• Paper 2, Section II, B

  #Differential Equations

  2015

  Write as a system of two first-order equations the second-order equation

  $$\frac{d^{2} \theta}{d t^{2}}+c \frac{d \theta}{d t}\left|\frac{d \theta}{d t}\right|+\sin \theta=0$$

  where $c$ is a small, positive constant, and find its equilibrium points. What is the nature of these points?

  Draw the trajectories in the $(\theta, \omega)$ plane, where $\omega=d \theta / d t$, in the neighbourhood of two typical equilibrium points.

  By considering the cases of $\omega>0$ and $\omega<0$ separately, find explicit expressions for $\omega^{2}$ as a function of $\theta$. Discuss how the second term in $(*)$ affects the nature of the equilibrium points.

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• Paper 2, Section II, B

  #Differential Equations

  2015

  Suppose that $u(x)$ satisfies the equation

  $$\frac{d^{2} u}{d x^{2}}-f(x) u=0$$

  where $f(x)$ is a given non-zero function. Show that under the change of coordinates $x=x(t)$,

  $$\frac{d^{2} u}{d t^{2}}-\frac{\ddot{x}}{\dot{x}} \frac{d u}{d t}-\dot{x}^{2} f(x) u=0$$

  where a dot denotes differentiation with respect to $t$. Furthermore, show that the function

  $$U(t)=\dot{x}^{-\frac{1}{2}} u(x)$$

  satisfies

  $$\frac{d^{2} U}{d t^{2}}-\left[\dot{x}^{2} f(x)+\dot{x}^{-\frac{1}{2}}\left(\frac{\ddot{x}}{\dot{x}} \frac{d}{d t}\left(\dot{x}^{\frac{1}{2}}\right)-\frac{d^{2}}{d t^{2}}\left(\dot{x}^{\frac{1}{2}}\right)\right)\right] U=0$$

  Choosing $\dot{x}=(f(x))^{-\frac{1}{2}}$, deduce that

  $$\frac{d^{2} U}{d t^{2}}-(1+F(t)) U=0$$

  for some appropriate function $F(t)$. Assuming that $F$ may be neglected, deduce that $u(x)$ can be approximated by

  $$u(x) \approx A(x)\left(c_{+} e^{G(x)}+c_{-} e^{-G(x)}\right),$$

  where $c_{+}, c_{-}$are constants and $A, G$ are functions that you should determine in terms of $f(x)$.

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• Paper 2, Section II, B

  #Differential Equations

  2015

  Suppose that $\mathbf{x}(t) \in \mathbb{R}^{3}$ obeys the differential equation

  $$\frac{d \mathbf{x}}{d t}=M \mathbf{x}$$

  where $M$ is a constant $3 \times 3$ real matrix.

  (i) Suppose that $M$ has distinct eigenvalues $\lambda_{1}, \lambda_{2}, \lambda_{3}$ with corresponding eigenvectors $\mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3}$. Explain why $\mathbf{x}$ may be expressed in the form $\sum_{i=1}^{3} a_{i}(t) \mathbf{e}_{i}$ and deduce by substitution that the general solution of $(*)$ is

  $$\mathbf{x}=\sum_{i=1}^{3} A_{i} e^{\lambda_{i} t} \mathbf{e}_{i}$$

  where $A_{1}, A_{2}, A_{3}$ are constants.

  (ii) What is the general solution of $(*)$ if $\lambda_{2}=\lambda_{3} \neq \lambda_{1}$, but there are still three linearly independent eigenvectors?

  (iii) Suppose again that $\lambda_{2}=\lambda_{3} \neq \lambda_{1}$, but now there are only two linearly independent eigenvectors: $\mathbf{e}_{1}$ corresponding to $\lambda_{1}$ and $\mathbf{e}_{2}$ corresponding to $\lambda_{2}$. Suppose that a vector $\mathbf{v}$ satisfying the equation $\left(M-\lambda_{2} I\right) \mathbf{v}=\mathbf{e}_{2}$ exists, where $I$ denotes the identity matrix. Show that $\mathbf{v}$ is linearly independent of $\mathbf{e}_{1}$ and $\mathbf{e}_{2}$, and hence or otherwise find the general solution of $(*)$.

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• Paper 4, Section I, C

  #Dynamics and Relativity

  2015

  Find the moment of inertia of a uniform sphere of mass $M$ and radius $a$ about an axis through its centre.

  The kinetic energy $T$ of any rigid body with total mass $M$, centre of mass $\mathbf{R}$, moment of inertia $I$ about an axis of rotation through $\mathbf{R}$, and angular velocity $\omega$ about that same axis, is given by $T=\frac{1}{2} M \dot{\mathbf{R}}^{2}+\frac{1}{2} I \omega^{2}$. What physical interpretation can be given to the two parts of this expression?

  A spherical marble of uniform density and mass $M$ rolls without slipping at speed $V$ along a flat surface. Explaining any relationship that you use between its speed and angular velocity, show that the kinetic energy of the marble is $\frac{7}{10} M V^{2}$.

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• Paper 2, Section II, B

  #Differential Equations

  2015

  Consider the equation

  $$2 \frac{\partial^{2} u}{\partial x^{2}}+3 \frac{\partial^{2} u}{\partial y^{2}}-7 \frac{\partial^{2} u}{\partial x \partial y}=0$$

  for the function $u(x, y)$, where $x$ and $y$ are real variables. By using the change of variables

  $$\xi=x+\alpha y, \quad \eta=\beta x+y$$

  where $\alpha$ and $\beta$ are appropriately chosen integers, transform $(*)$ into the equation

  $$\frac{\partial^{2} u}{\partial \xi \partial \eta}=0$$

  Hence, solve equation $(*)$ supplemented with the boundary conditions

  $$u(0, y)=4 y^{2}, \quad u(-2 y, y)=0, \quad \text { for all } y$$

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• Paper 4, Section I, C

  #Dynamics and Relativity

  2015

  Write down the 4-momentum of a particle with energy $E$ and 3-momentum p. State the relationship between the energy $E$ and wavelength $\lambda$ of a photon.

  An electron of mass $m$ is at rest at the origin of the laboratory frame: write down its 4 -momentum. The electron is scattered by a photon of wavelength $\lambda_{1}$ travelling along the $x$-axis: write down the initial 4-momentum of the photon. Afterwards, the photon has wavelength $\lambda_{2}$ and has been deflected through an angle $\theta$. Show that

  $$\lambda_{2}-\lambda_{1}=\frac{2 h}{m c} \sin ^{2}\left(\frac{1}{2} \theta\right)$$

  where $c$ is the speed of light and $h$ is Planck's constant.

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• Paper 4, Section II, $\mathbf{9 C}$

  #Dynamics and Relativity

  2015

  A particle is projected vertically upwards at speed $V$ from the surface of the Earth, which may be treated as a perfect sphere. The variation of gravity with height should not be ignored, but the rotation of the Earth should be. Show that the height $z(t)$ of the particle obeys

  $$\ddot{z}=-\frac{g R^{2}}{(R+z)^{2}},$$

  where $R$ is the radius of the Earth and $g$ is the acceleration due to gravity measured at the Earth's surface.

  Using dimensional analysis, show that the maximum height $H$ of the particle and the time $T$ taken to reach that height are given by

  $$H=R F(\lambda) \quad \text { and } \quad T=\frac{V}{g} G(\lambda)$$

  where $F$ and $G$ are functions of $\lambda=V^{2} / g R$.

  Write down the equation of conservation of energy and deduce that

  $$T=\int_{0}^{H} \sqrt{\frac{R+z}{V^{2} R-\left(2 g R-V^{2}\right) z}} d z$$

  Hence or otherwise show that

  $$F(\lambda)=\frac{\lambda}{2-\lambda} \quad \text { and } \quad G(\lambda)=\int_{0}^{1} \sqrt{\frac{2-\lambda+\lambda x}{(2-\lambda)^{3}(1-x)}} d x$$

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• Paper 4, Section II, C

  #Dynamics and Relativity

  2015

  A particle of mass $m$ and charge $q$ has position vector $\mathbf{r}(t)$ and moves in a constant, uniform magnetic field $\mathbf{B}$ so that its equation of motion is

  $$m \ddot{\mathbf{r}}=q \dot{\mathbf{r}} \times \mathbf{B}$$

  Let $\mathbf{L}=m \mathbf{r} \times \dot{\mathbf{r}}$ be the particle's angular momentum. Show that

  $$\mathbf{L} \cdot \mathbf{B}+\frac{1}{2} q|\mathbf{r} \times \mathbf{B}|^{2}$$

  is a constant of the motion. Explain why the kinetic energy $T$ is also constant, and show that it may be written in the form

  $$T=\frac{1}{2} m \mathbf{u} \cdot\left((\mathbf{u} \cdot \mathbf{v}) \mathbf{v}-r^{2} \ddot{\mathbf{u}}\right)$$

  where $\mathbf{v}=\dot{\mathbf{r}}, r=|\mathbf{r}|$ and $\mathbf{u}=\mathbf{r} / r$.

  [Hint: Consider u $\cdot \dot{\mathbf{u}} .]$

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• Paper 4, Section II, C

  #Dynamics and Relativity

  2015

  Consider a particle with position vector $r(t)$ moving in a plane described by polar coordinates $(r, \theta)$. Obtain expressions for the radial $(r)$ and transverse $(\theta)$ components of the velocity $\dot{\mathbf{r}}$ and acceleration $\ddot{\mathbf{r}}$.

  A charged particle of unit mass moves in the electric field of another charge that is fixed at the origin. The electrostatic force on the particle is $-p / r^{2}$ in the radial direction, where $p$ is a positive constant. The motion takes place in an unusual medium that resists radial motion but not tangential motion, so there is an additional radial force $-k \dot{r} / r^{2}$ where $k$ is a positive constant. Show that the particle's motion lies in a plane. Using polar coordinates in that plane, show also that its angular momentum $h=r^{2} \dot{\theta}$ is constant.

  Obtain the equation of motion

  $$\frac{d^{2} u}{d \theta^{2}}+\frac{k}{h} \frac{d u}{d \theta}+u=\frac{p}{h^{2}}$$

  where $u=r^{-1}$, and find its general solution assuming that $k /|h|<2$. Show that so long as the motion remains bounded it eventually becomes circular with radius $h^{2} / p$.

  Obtain the expression

  $$E=\frac{1}{2} h^{2}\left(u^{2}+\left(\frac{d u}{d \theta}\right)^{2}\right)-p u$$

  for the particle's total energy, that is, its kinetic energy plus its electrostatic potential energy. Hence, or otherwise, show that the energy is a decreasing function of time.

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• Paper 4, Section II, C

  #Dynamics and Relativity

  2015

  Write down the Lorentz transform relating the components of a 4-vector between two inertial frames.

  A particle moves along the $x$-axis of an inertial frame. Its position at time $t$ is $x(t)$, its velocity is $u=d x / d t$, and its 4 -position is $X=(c t, x)$, where $c$ is the speed of light. The particle's 4-velocity is given by $U=d X / d \tau$ and its 4 -acceleration is $A=d U / d \tau$, where proper time $\tau$ is defined by $c^{2} d \tau^{2}=c^{2} d t^{2}-d x^{2}$. Show that

  $$U=\gamma(c, u) \quad \text { and } \quad A=\gamma^{4} \dot{u}(u / c, 1)$$

  where $\gamma=\left(1-u^{2} / c^{2}\right)^{-\frac{1}{2}}$ and $\dot{u}=d u / d t$.

  The proper 3-acceleration a of the particle is defined to be the spatial component of its 4-acceleration measured in the particle's instantaneous rest frame. By transforming $A$ to the rest frame, or otherwise, show that

  $$a=\gamma^{3} \dot{u}=\frac{d}{d t}(\gamma u)$$

  Given that the particle moves with constant proper 3 -acceleration starting from rest at the origin, show that

  $$x(t)=\frac{c^{2}}{a}\left(\sqrt{1+\frac{a^{2} t^{2}}{c^{2}}}-1\right)$$

  and that, if $a t \ll c$, then $x \approx \frac{1}{2} a t^{2}$.

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• Paper 3, Section I, D

  #Groups

  2015

  Say that a group is dihedral if it has two generators $x$ and $y$, such that $x$ has order $n$ (greater than or equal to 2 and possibly infinite), $y$ has order 2 , and $y x y^{-1}=x^{-1}$. In particular the groups $C_{2}$ and $C_{2} \times C_{2}$ are regarded as dihedral groups. Prove that:

  (i) any dihedral group can be generated by two elements of order 2 ;

  (ii) any group generated by two elements of order 2 is dihedral; and

  (iii) any non-trivial quotient group of a dihedral group is dihedral.

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• Paper 3, Section I, D

  #Groups

  2015

  How many cyclic subgroups (including the trivial subgroup) does $S_{5}$ contain? Exhibit two isomorphic subgroups of $S_{5}$ which are not conjugate.

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• Paper 3, Section II, D

  #Groups

  2015

  What does it mean for a group $G$ to act on a set $X$ ? For $x \in X$, what is meant by the orbit $\operatorname{Orb}(x)$ to which $x$ belongs, and by the stabiliser $G_{x}$ of $x$ ? Show that $G_{x}$ is a subgroup of $G$. Prove that, if $G$ is finite, then $|G|=\left|G_{x}\right| \cdot|\operatorname{Orb}(x)|$.

  (a) Prove that the symmetric group $S_{n}$ acts on the set $P^{(n)}$ of all polynomials in $n$ variables $x_{1}, \ldots, x_{n}$, if we define $\sigma \cdot f$ to be the polynomial given by

  $$(\sigma \cdot f)\left(x_{1}, \ldots, x_{n}\right)=f\left(x_{\sigma(1)}, \ldots, x_{\sigma(n)}\right)$$

  for $f \in P^{(n)}$ and $\sigma \in S_{n}$. Find the orbit of $f=x_{1} x_{2}+x_{3} x_{4} \in P^{(4)}$ under $S_{4}$. Find also the order of the stabiliser of $f$.

  (b) Let $r, n$ be fixed positive integers such that $r \leqslant n$. Let $B_{r}$ be the set of all subsets of size $r$ of the set $\{1,2, \ldots, n\}$. Show that $S_{n}$ acts on $B_{r}$ by defining $\sigma \cdot U$ to be the set $\{\sigma(u): u \in U\}$, for any $U \in B_{r}$ and $\sigma \in S_{n}$. Prove that $S_{n}$ is transitive in its action on $B_{r}$. Find also the size of the stabiliser of $U \in B_{r}$.

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• Paper 3, Section II, D

  #Groups

  2015

  Let $G, H$ be groups and let $\varphi: G \rightarrow H$ be a function. What does it mean to say that $\varphi$ is a homomorphism with kernel $K$ ? Show that if $K=\{e, \xi\}$ has order 2 then $x^{-1} \xi x=\xi$ for each $x \in G$. [If you use any general results about kernels of homomorphisms, then you should prove them.]

  Which of the following four statements are true, and which are false? Justify your answers.

  (a) There is a homomorphism from the orthogonal group $\mathrm{O}(3)$ to a group of order 2 with kernel the special orthogonal group $\mathrm{SO}(3)$.

  (b) There is a homomorphism from the symmetry group $S_{3}$ of an equilateral triangle to a group of order 2 with kernel of order 3 .

  (c) There is a homomorphism from $\mathrm{O}(3)$ to $\mathrm{SO}(3)$ with kernel of order 2 .

  (d) There is a homomorphism from $S_{3}$ to a group of order 3 with kernel of order 2 .

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• Paper 3, Section II, D

  #Groups

  2015

  (a) State and prove Lagrange's theorem.

  (b) Let $G$ be a group and let $H, K$ be fixed subgroups of $G$. For each $g \in G$, any set of the form $H g K=\{h g k: h \in H, k \in K\}$ is called an $(H, K)$ double coset, or simply a double coset if $H$ and $K$ are understood. Prove that every element of $G$ lies in some $(H, K)$ double coset, and that any two $(H, K)$ double cosets either coincide or are disjoint.

  Let $G$ be a finite group. Which of the following three statements are true, and which are false? Justify your answers.

  (i) The size of a double coset divides the order of $G$.

  (ii) Different double cosets for the same pair of subgroups have the same size.

  (iii) The number of double cosets divides the order of $G$.

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• Paper 3, Section II, D

  #Groups

  2015

  (a) Let $G$ be a non-trivial group and let $Z(G)=\{h \in G: g h=h g$ for all $g \in G\}$. Show that $Z(G)$ is a normal subgroup of $G$. If the order of $G$ is a power of a prime, show that $Z(G)$ is non-trivial.

  (b) The Heisenberg group $H$ is the set of all $3 \times 3$ matrices of the form

  $$\left(\begin{array}{lll} 1 & x & y \\ 0 & 1 & z \\ 0 & 0 & 1 \end{array}\right)$$

  with $x, y, z \in \mathbb{R}$. Show that $H$ is a subgroup of the group of non-singular real matrices under matrix multiplication.

  Find $Z(H)$ and show that $H / Z(H)$ is isomorphic to $\mathbb{R}^{2}$ under vector addition.

  (c) For $p$ prime, the modular Heisenberg group $H_{p}$ is defined as in (b), except that $x, y$ and $z$ now lie in the field of $p$ elements. Write down $\left|H_{p}\right|$. Find both $Z\left(H_{p}\right)$ and $H_{p} / Z\left(H_{p}\right)$ in terms of generators and relations.

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• Paper 4, Section I, E

  #Numbers and Sets

  2015

  (a) Find all integers $x$ and $y$ such that

  $$6 x+2 y \equiv 3 \quad(\bmod 53) \quad \text { and } \quad 17 x+4 y \equiv 7 \quad(\bmod 53)$$

  (b) Show that if an integer $n>4$ is composite then $(n-1) ! \equiv 0(\bmod n)$.

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• Paper 4 , Section I, E

  #Numbers and Sets

  2015

  State the Chinese remainder theorem and Fermat's theorem. Prove that

  $$p^{4} \equiv 1 \quad(\bmod 240)$$

  for any prime $p>5$.

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• Paper 4, Section II, E

  #Numbers and Sets

  2015

  (i) Let $\sim$ be an equivalence relation on a set $X$. What is an equivalence class of $\sim$ ? What is a partition of $X ?$ Prove that the equivalence classes of $\sim$ form a partition of $X$.

  (ii) Let $\sim$ be the relation on the natural numbers $\mathbb{N}=\{1,2,3, \ldots\}$ defined by

  $$m \sim n \Longleftrightarrow \exists a, b \in \mathbb{N} \text { such that } m \text { divides } n^{a} \text { and } n \text { divides } m^{b} .$$

  Show that $\sim$ is an equivalence relation, and show that it has infinitely many equivalence classes, all but one of which are infinite.

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• Paper 4, Section II, E

  #Numbers and Sets

  2015

  Let $p$ be a prime. A base $p$ expansion of an integer $k$ is an expression

  $$k=k_{0}+p \cdot k_{1}+p^{2} \cdot k_{2}+\cdots+p^{\ell} \cdot k_{\ell}$$

  for some natural number $\ell$, with $0 \leqslant k_{i}<p$ for $i=0,1, \ldots, \ell$.

  (i) Show that the sequence of coefficients $k_{0}, k_{1}, k_{2}, \ldots, k_{\ell}$ appearing in a base $p$ expansion of $k$ is unique, up to extending the sequence by zeroes.

  (ii) Show that

  $$\left(\begin{array}{l} p \\ j \end{array}\right) \equiv 0 \quad(\bmod p), \quad 0<j<p$$

  and hence, by considering the polynomial $(1+x)^{p}$ or otherwise, deduce that

  $$\left(\begin{array}{c} p^{i} \\ j \end{array}\right) \equiv 0 \quad(\bmod p), \quad 0<j<p^{i}$$

  (iii) If $n_{0}+p \cdot n_{1}+p^{2} \cdot n_{2}+\cdots+p^{\ell} \cdot n_{\ell}$ is a base $p$ expansion of $n$, then, by considering the polynomial $(1+x)^{n}$ or otherwise, show that

  $$\left(\begin{array}{l} n \\ k \end{array}\right) \equiv\left(\begin{array}{l} n_{0} \\ k_{0} \end{array}\right)\left(\begin{array}{l} n_{1} \\ k_{1} \end{array}\right) \cdots\left(\begin{array}{l} n_{\ell} \\ k_{\ell} \end{array}\right) \quad(\bmod p)$$

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• Paper 4, Section II, E

  #Numbers and Sets

  2015

  State the inclusion-exclusion principle.

  Let $n \in \mathbb{N}$. A permutation $\sigma$ of the set $\{1,2,3, \ldots, n\}$ is said to contain a transposition if there exist $i, j$ with $1 \leqslant i<j \leqslant n$ such that $\sigma(i)=j$ and $\sigma(j)=i$. Derive a formula for the number, $f(n)$, of permutations which do not contain a transposition, and show that

  $$\lim _{n \rightarrow \infty} \frac{f(n)}{n !}=e^{-\frac{1}{2}}$$

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• Paper 4, Section II, E

  #Numbers and Sets

  2015

  What does it mean for a set to be countable? Prove that

  (a) if $B$ is countable and $f: A \rightarrow B$ is injective, then $A$ is countable;

  (b) if $A$ is countable and $f: A \rightarrow B$ is surjective, then $B$ is countable.

  Prove that $\mathbb{N} \times \mathbb{N}$ is countable, and deduce that

  (i) if $X$ and $Y$ are countable, then so is $X \times Y$;

  (ii) $\mathbb{Q}$ is countable.

  Let $\mathcal{C}$ be a collection of circles in the plane such that for each point $a$ on the $x$-axis, there is a circle in $\mathcal{C}$ passing through the point $a$ which has the $x$-axis tangent to the circle at $a$. Show that $\mathcal{C}$ contains a pair of circles that intersect.

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• Paper 2, Section I, F

  #Probability

  2015

  Let $U$ be a uniform random variable on $(0,1)$, and let $\lambda>0$.

  (a) Find the distribution of the random variable $-(\log U) / \lambda$.

  (b) Define a new random variable $X$ as follows: suppose a fair coin is tossed, and if it lands heads we set $X=U^{2}$ whereas if it lands tails we set $X=1-U^{2}$. Find the probability density function of $X$.

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• Paper 2, Section I, F

  #Probability

  2015

  Let $A, B$ be events in the sample space $\Omega$ such that $0<P(A)<1$ and $0<P(B)<1$. The event $B$ is said to attract $A$ if the conditional probability $P(A \mid B)$ is greater than $P(A)$, otherwise it is said that $A$ repels $B$. Show that if $B$ attracts $A$, then $A$ attracts $B$. Does $B^{c}=\Omega \backslash B$ repel $A ?$

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• Paper 2, Section II, F

  #Probability

  2015

  Lionel and Cristiana have $a$ and $b$ million pounds, respectively, where $a, b \in \mathbb{N}$. They play a series of independent football games in each of which the winner receives one million pounds from the loser (a draw cannot occur). They stop when one player has lost his or her entire fortune. Lionel wins each game with probability $0<p<1$ and Cristiana wins with probability $q=1-p$, where $p \neq q$. Find the expected number of games before they stop playing.

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• Paper 2, Section II, F

  #Probability

  2015

  Consider the function

  $$\phi(x)=\frac{1}{\sqrt{2 \pi}} e^{-x^{2} / 2}, \quad x \in \mathbb{R}$$

  Show that $\phi$ defines a probability density function. If a random variable $X$ has probability density function $\phi$, find the moment generating function of $X$, and find all moments $E\left[X^{k}\right]$, $k \in \mathbb{N}$.

  Now define

  $$r(x)=\frac{P(X>x)}{\phi(x)}$$

  Show that for every $x>0$,

  $$\frac{1}{x}-\frac{1}{x^{3}}<r(x)<\frac{1}{x}$$

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• Paper 2, Section II, F

  #Probability

  2015

  State and prove Markov's inequality and Chebyshev's inequality, and deduce the weak law of large numbers.

  If $X$ is a random variable with mean zero and finite variance $\sigma^{2}$, prove that for any $a>0$,

  $$P(X \geqslant a) \leqslant \frac{\sigma^{2}}{\sigma^{2}+a^{2}}$$

  [Hint: Show first that $P(X \geqslant a) \leqslant P\left((X+b)^{2} \geqslant(a+b)^{2}\right)$ for every $b>0$.]

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• Paper 2, Section II, F

  #Probability

  2015

  When coin $A$ is tossed it comes up heads with probability $\frac{1}{4}$, whereas coin $B$ comes up heads with probability $\frac{3}{4}$. Suppose one of these coins is randomly chosen and is tossed twice. If both tosses come up heads, what is the probability that coin $B$ was tossed? Justify your answer.

  In each draw of a lottery, an integer is picked independently at random from the first $n$ integers $1,2, \ldots, n$, with replacement. What is the probability that in a sample of $r$ successive draws the numbers are drawn in a non-decreasing sequence? Justify your answer.

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• Paper 3, Section I, A

  #Vector Calculus

  2015

  (i) For $r=|\mathbf{x}|$ with $\mathbf{x} \in \mathbb{R}^{3} \backslash\{\mathbf{0}\}$, show that

  $$\frac{\partial r}{\partial x_{i}}=\frac{x_{i}}{r} \quad(i=1,2,3) .$$

  (ii) Consider the vector fields $\mathbf{F}(\mathbf{x})=r^{2} \mathbf{x}, \mathbf{G}(\mathbf{x})=(\mathbf{a} \cdot \mathbf{x}) \mathbf{x}$ and $\mathbf{H}(\mathbf{x})=\mathbf{a} \times \hat{\mathbf{x}}$, where $\mathbf{a}$ is a constant vector in $\mathbb{R}^{3}$ and $\hat{\mathbf{x}}$ is the unit vector in the direction of $\mathbf{x}$. Using suffix notation, or otherwise, find the divergence and the curl of each of $\mathbf{F}, \mathbf{G}$ and $\mathbf{H}$.

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• Paper 3, Section I, A

  #Vector Calculus

  2015

  The smooth curve $\mathcal{C}$ in $\mathbb{R}^{3}$ is given in parametrised form by the function $\mathbf{x}(u)$. Let $s$ denote arc length measured along the curve.

  (a) Express the tangent $\mathbf{t}$ in terms of the derivative $\mathbf{x}^{\prime}=d \mathbf{x} / d u$, and show that $d u / d s=\left|\mathbf{x}^{\prime}\right|^{-1}$.

  (b) Find an expression for $d \mathbf{t} / d s$ in terms of derivatives of $\mathbf{x}$ with respect to $u$, and show that the curvature $\kappa$ is given by

  $$\kappa=\frac{\left|\mathbf{x}^{\prime} \times \mathbf{x}^{\prime \prime}\right|}{\left|\mathbf{x}^{\prime}\right|^{3}}$$

  [Hint: You may find the identity $\left(\mathbf{x}^{\prime} \cdot \mathbf{x}^{\prime \prime}\right) \mathbf{x}^{\prime}-\left(\mathbf{x}^{\prime} \cdot \mathbf{x}^{\prime}\right) \mathbf{x}^{\prime \prime}=\mathbf{x}^{\prime} \times\left(\mathbf{x}^{\prime} \times \mathbf{x}^{\prime \prime}\right)$ helpful.]

  (c) For the curve

  $$\mathbf{x}(u)=\left(\begin{array}{c} u \cos u \\ u \sin u \\ 0 \end{array}\right)$$

  with $u \geqslant 0$, find the curvature as a function of $u$.

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• Paper 3, Section II, A

  #Vector Calculus

  2015

  The vector field $\mathbf{F}(\mathbf{x})$ is given in terms of cylindrical polar coordinates $(\rho, \phi, z)$ by

  $$\mathbf{F}(\mathbf{x})=f(\rho) \mathbf{e}_{\rho}$$

  where $f$ is a differentiable function of $\rho$, and $\mathbf{e}_{\rho}=\cos \phi \mathbf{e}_{x}+\sin \phi \mathbf{e}_{y}$ is the unit basis vector with respect to the coordinate $\rho$. Compute the partial derivatives $\partial F_{1} / \partial x, \partial F_{2} / \partial y$, $\partial F_{3} / \partial z$ and hence find the divergence $\nabla \cdot \mathbf{F}$ in terms of $\rho$ and $\phi$.

  The domain $V$ is bounded by the surface $z=\left(x^{2}+y^{2}\right)^{-1}$, by the cylinder $x^{2}+y^{2}=1$, and by the planes $z=\frac{1}{4}$ and $z=1$. Sketch $V$ and compute its volume.

  Find the most general function $f(\rho)$ such that $\nabla \cdot \mathbf{F}=0$, and verify the divergence theorem for the corresponding vector field $\mathbf{F}(\mathbf{x})$ in $V$.

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• Paper 3, Section II, A

  #Vector Calculus

  2015

  State Stokes' theorem.

  Let $S$ be the surface in $\mathbb{R}^{3}$ given by $z^{2}=x^{2}+y^{2}+1-\lambda$, where $0 \leqslant z \leqslant 1$ and $\lambda$ is a positive constant. Sketch the surface $S$ for representative values of $\lambda$ and find the surface element $\mathbf{d} \mathbf{S}$ with respect to the Cartesian coordinates $x$ and $y$.

  Compute $\nabla \times \mathbf{F}$ for the vector field

  $$\mathbf{F}(\mathbf{x})=\left(\begin{array}{c} -y \\ x \\ z \end{array}\right)$$

  and verify Stokes' theorem for $\mathbf{F}$ on the surface $S$ for every value of $\lambda$.

  Now compute $\nabla \times \mathbf{G}$ for the vector field

  $$\mathbf{G}(\mathbf{x})=\frac{1}{x^{2}+y^{2}}\left(\begin{array}{c} -y \\ x \\ 0 \end{array}\right)$$

  and find the line integral $\int_{\partial S} \mathbf{G} \cdot \mathbf{d x}$ for the boundary $\partial S$ of the surface $S$. Is it possible to obtain this result using Stokes' theorem? Justify your answer.

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• Paper 3, Section II, A

  #Vector Calculus

  2015

  (i) Starting with the divergence theorem, derive Green's first theorem

  $$\int_{V}\left(\psi \nabla^{2} \phi+\nabla \psi \cdot \nabla \phi\right) d V=\int_{\partial V} \psi \frac{\partial \phi}{\partial n} d S$$

  (ii) The function $\phi(\mathbf{x})$ satisfies Laplace's equation $\nabla^{2} \phi=0$ in the volume $V$ with given boundary conditions $\phi(\mathbf{x})=g(\mathbf{x})$ for all $\mathbf{x} \in \partial V$. Show that $\phi(\mathbf{x})$ is the only such function. Deduce that if $\phi(\mathbf{x})$ is constant on $\partial V$ then it is constant in the whole volume $V$.

  (iii) Suppose that $\phi(\mathbf{x})$ satisfies Laplace's equation in the volume $V$. Let $V_{r}$ be the sphere of radius $r$ centred at the origin and contained in $V$. The function $f(r)$ is defined by

  $$f(r)=\frac{1}{4 \pi r^{2}} \int_{\partial V_{r}} \phi(\mathbf{x}) d S$$

  By considering the derivative $d f / d r$, and by introducing the Jacobian in spherical polar coordinates and using the divergence theorem, or otherwise, show that $f(r)$ is constant and that $f(r)=\phi(\mathbf{0})$.

  (iv) Let $M$ denote the maximum of $\phi$ on $\partial V_{r}$ and $m$ the minimum of $\phi$ on $\partial V_{r}$. By using the result from (iii), or otherwise, show that $m \leqslant \phi(\mathbf{0}) \leqslant M$.

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• Paper 3, Section II, A

  #Vector Calculus

  2015

  (a) Let $t_{i j}$ be a rank 2 tensor whose components are invariant under rotations through an angle $\pi$ about each of the three coordinate axes. Show that $t_{i j}$ is diagonal.

  (b) An array of numbers $a_{i j}$ is given in one orthonormal basis as $\delta_{i j}+\epsilon_{1 i j}$ and in another rotated basis as $\delta_{i j}$. By using the invariance of the determinant of any rank 2 tensor, or otherwise, prove that $a_{i j}$ is not a tensor.

  (c) Let $a_{i j}$ be an array of numbers and $b_{i j}$ a tensor. Determine whether the following statements are true or false. Justify your answers.

  (i) If $a_{i j} b_{i j}$ is a scalar for any rank 2 tensor $b_{i j}$, then $a_{i j}$ is a rank 2 tensor.

  (ii) If $a_{i j} b_{i j}$ is a scalar for any symmetric rank 2 tensor $b_{i j}$, then $a_{i j}$ is a rank 2 tensor.

  (iii) If $a_{i j}$ is antisymmetric and $a_{i j} b_{i j}$ is a scalar for any symmetric rank 2 tensor $b_{i j}$, then $a_{i j}$ is an antisymmetric rank 2 tensor.

  (iv) If $a_{i j}$ is antisymmetric and $a_{i j} b_{i j}$ is a scalar for any antisymmetric rank 2 tensor $b_{i j}$, then $a_{i j}$ is an antisymmetric rank 2 tensor.

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• Paper 1, Section I, B

  #Vectors and Matrices

  2015

  (a) Describe geometrically the curve

  $$|\alpha z+\beta \bar{z}|=\sqrt{\alpha \beta}(z+\bar{z})+(\alpha-\beta)^{2},$$

  where $z \in \mathbb{C}$ and $\alpha, \beta$ are positive, distinct, real constants.

  (b) Let $\theta$ be a real number not equal to an integer multiple of $2 \pi$. Show that

  $$\sum_{m=1}^{N} \sin (m \theta)=\frac{\sin \theta+\sin (N \theta)-\sin (N \theta+\theta)}{2(1-\cos \theta)}$$

  and derive a similar expression for $\sum_{m=1}^{N} \cos (m \theta)$.

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• Paper 1, Section I, $2 \mathrm{C}$

  #Vectors and Matrices

  2015

  Precisely one of the four matrices specified below is not orthogonal. Which is it?

  Give a brief justification.

  $$\frac{1}{\sqrt{6}}\left(\begin{array}{rcc} 1 & -\sqrt{3} & \sqrt{2} \\ 1 & \sqrt{3} & \sqrt{2} \\ -2 & 0 & \sqrt{2} \end{array}\right) \quad \frac{1}{3}\left(\begin{array}{ccc} 1 & 2 & -2 \\ 2 & -2 & -1 \\ 2 & 1 & 2 \end{array}\right) \quad \frac{1}{\sqrt{6}}\left(\begin{array}{rrr} 1 & -2 & 1 \\ -\sqrt{6} & 0 & \sqrt{6} \\ 1 & 1 & 1 \end{array}\right) \quad \frac{1}{9}\left(\begin{array}{rrr} 7 & -4 & -4 \\ -4 & 1 & -8 \\ -4 & -8 & 1 \end{array}\right)$$

  Given that the four matrices represent transformations of $\mathbb{R}^{3}$ corresponding (in no particular order) to a rotation, a reflection, a combination of a rotation and a reflection, and none of these, identify each matrix. Explain your reasoning.

  [Hint: For two of the matrices, $A$ and $B$ say, you may find it helpful to calculate $\operatorname{det}(A-I)$ and $\operatorname{det}(B-I)$, where $I$ is the identity matrix.]

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• Paper 1, Section II, 5B

  #Vectors and Matrices

  2015

  (i) State and prove the Cauchy-Schwarz inequality for vectors in $\mathbb{R}^{n}$. Deduce the inequalities

  $$|\mathbf{a}+\mathbf{b}| \leqslant|\mathbf{a}|+|\mathbf{b}| \text { and }|\mathbf{a}+\mathbf{b}+\mathbf{c}| \leqslant|\mathbf{a}|+|\mathbf{b}|+|\mathbf{c}|$$

  for $\mathbf{a}, \mathbf{b}, \mathbf{c} \in \mathbb{R}^{n}$.

  (ii) Show that every point on the intersection of the planes

  $$\mathbf{x} \cdot \mathbf{a}=A, \quad \mathbf{x} \cdot \mathbf{b}=B$$

  where $\mathbf{a} \neq \mathbf{b}$, satisfies

  $$|\mathbf{x}|^{2} \geqslant \frac{(A-B)^{2}}{|\mathbf{a}-\mathbf{b}|^{2}}$$

  What happens if $\mathbf{a}=\mathbf{b} ?$

  (iii) Using your results from part (i), or otherwise, show that for any $\mathbf{x}_{1}, \mathbf{x}_{2}, \mathbf{y}_{1}, \mathbf{y}_{2} \in \mathbb{R}^{n}$,

  $$\left|\mathbf{x}_{1}-\mathbf{y}_{1}\right|-\left|\mathbf{x}_{1}-\mathbf{y}_{2}\right| \leqslant\left|\mathbf{x}_{2}-\mathbf{y}_{1}\right|+\left|\mathbf{x}_{2}-\mathbf{y}_{2}\right|$$

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• Paper 1, Section II, $6 \mathrm{C}$

  #Vectors and Matrices

  2015

  (i) Consider the map from $\mathbb{R}^{4}$ to $\mathbb{R}^{3}$ represented by the matrix

  $$\left(\begin{array}{rrrr} \alpha & 1 & 1 & -1 \\ 2 & -\alpha & 0 & -2 \\ -\alpha & 2 & 1 & 1 \end{array}\right)$$

  where $\alpha \in \mathbb{R}$. Find the image and kernel of the map for each value of $\alpha$.

  (ii) Show that any linear map $f: \mathbb{R}^{n} \rightarrow \mathbb{R}$ may be written in the form $f(\mathbf{x})=\mathbf{a} \cdot \mathbf{x}$ for some fixed vector $\mathbf{a} \in \mathbb{R}^{n}$. Show further that $\mathbf{a}$ is uniquely determined by $f$.

  It is given that $n=4$ and that the vectors

  $$\left(\begin{array}{r} 1 \\ 1 \\ 1 \\ -1 \end{array}\right),\left(\begin{array}{r} 2 \\ -1 \\ 0 \\ -2 \end{array}\right),\left(\begin{array}{r} -1 \\ 2 \\ 1 \\ 1 \end{array}\right)$$

  lie in the kernel of $f$. Determine the set of possible values of a.

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• Paper 1, Section II, A

  #Vectors and Matrices

  2015

  (i) Find the eigenvalues and eigenvectors of the following matrices and show that both are diagonalisable:

  $$A=\left(\begin{array}{rrr} 1 & 1 & -1 \\ -1 & 3 & -1 \\ -1 & 1 & 1 \end{array}\right), \quad B=\left(\begin{array}{rcr} 1 & 4 & -3 \\ -4 & 10 & -4 \\ -3 & 4 & 1 \end{array}\right)$$

  (ii) Show that, if two real $n \times n$ matrices can both be diagonalised using the same basis transformation, then they commute.

  (iii) Suppose now that two real $n \times n$ matrices $C$ and $D$ commute and that $D$ has $n$ distinct eigenvalues. Show that for any eigenvector $\mathbf{x}$ of $D$ the vector $C \mathbf{x}$ is a scalar multiple of $\mathbf{x}$. Deduce that there exists a common basis transformation that diagonalises both matrices.

  (iv) Show that $A$ and $B$ satisfy the conditions in (iii) and find a matrix $S$ such that both of the matrices $S^{-1} A S$ and $S^{-1} B S$ are diagonal.

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• Paper 1, Section II, A

  #Vectors and Matrices

  2015

  (a) A matrix is called normal if $A^{\dagger} A=A A^{\dagger}$. Let $A$ be a normal $n \times n$ complex matrix.

  (i) Show that for any vector $\mathbf{x} \in \mathbb{C}^{n}$,

  $$|A \mathbf{x}|=\left|A^{\dagger} \mathbf{x}\right|$$

  (ii) Show that $A-\lambda I$ is also normal for any $\lambda \in \mathbb{C}$, where $I$ denotes the identity matrix.

  (iii) Show that if $\mathbf{x}$ is an eigenvector of $A$ with respect to the eigenvalue $\lambda \in \mathbb{C}$, then $\mathbf{x}$ is also an eigenvector of $A^{\dagger}$, and determine the corresponding eigenvalue.

  (iv) Show that if $\mathbf{x}_{\lambda}$ and $\mathbf{x}_{\mu}$ are eigenvectors of $A$ with respect to distinct eigenvalues $\lambda$ and $\mu$ respectively, then $\mathbf{x}_{\lambda}$ and $\mathbf{x}_{\mu}$ are orthogonal.

  (v) Show that if $A$ has a basis of eigenvectors, then $A$ can be diagonalised using an orthonormal basis. Justify your answer.

  [You may use standard results provided that they are clearly stated.]

  (b) Show that any matrix $A$ satisfying $A^{\dagger}=A$ is normal, and deduce using results from (a) that its eigenvalues are real.

  (c) Show that any matrix $A$ satisfying $A^{\dagger}=-A$ is normal, and deduce using results from (a) that its eigenvalues are purely imaginary.

  (d) Show that any matrix $A$ satisfying $A^{\dagger}=A^{-1}$ is normal, and deduce using results from (a) that its eigenvalues have unit modulus.

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• Paper 3, Section I, G

  #Analysis II

  2015

  Define what is meant by a uniformly continuous function $f$ on a subset $E$ of a metric space. Show that every continuous function on a closed, bounded interval is uniformly continuous. [You may assume the Bolzano-Weierstrass theorem.]

  Suppose that a function $g:[0, \infty) \rightarrow \mathbb{R}$ is continuous and tends to a finite limit at $\infty$. Is $g$ necessarily uniformly continuous on $[0, \infty) ?$ Give a proof or a counterexample as appropriate.

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• Paper 4, Section I, G

  #Analysis II

  2015

  Define what is meant for two norms on a vector space to be Lipschitz equivalent.

  Let $C_{c}^{1}([-1,1])$ denote the vector space of continuous functions $f:[-1,1] \rightarrow \mathbb{R}$ with continuous first derivatives and such that $f(x)=0$ for $x$ in some neighbourhood of the end-points $-1$ and 1 . Which of the following four functions $C_{c}^{1}([-1,1]) \rightarrow \mathbb{R}$ define norms on $C_{c}^{1}([-1,1])$ (give a brief explanation)?

  Among those that define norms, which pairs are Lipschitz equivalent? Justify your answer.

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• Paper 2, Section I, G

  #Analysis II

  2015

  Show that the map $f: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$ given by

  $$f(x, y, z)=\left(x-y-z, x^{2}+y^{2}+z^{2}, x y z\right)$$

  is differentiable everywhere and find its derivative.

  Stating accurately any theorem that you require, show that $f$ has a differentiable local inverse at a point $(x, y, z)$ if and only if

  $$(x+y)(x+z)(y-z) \neq 0 .$$

  $$\begin{aligned} & p(f)=\sup |f|, \quad q(f)=\sup \left(|f|+\left|f^{\prime}\right|\right), \\ & r(f)=\sup \left|f^{\prime}\right|, \quad s(f)=\left|\int_{-1}^{1} f(x) d x\right| \end{aligned}$$

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• Paper 1, Section II, G

  #Analysis II

  2015

  Define what it means for a sequence of functions $f_{n}:[0,1] \rightarrow \mathbb{R}$ to converge uniformly on $[0,1]$ to a function $f$.

  Let $f_{n}(x)=n^{p} x e^{-n^{q} x}$, where $p, q$ are positive constants. Determine all the values of $(p, q)$ for which $f_{n}(x)$ converges pointwise on $[0,1]$. Determine all the values of $(p, q)$ for which $f_{n}(x)$ converges uniformly on $[0,1]$.

  Let now $f_{n}(x)=e^{-n x^{2}}$. Determine whether or not $f_{n}$ converges uniformly on $[0,1]$.

  Let $f:[0,1] \rightarrow \mathbb{R}$ be a continuous function. Show that the sequence $x^{n} f(x)$ is uniformly convergent on $[0,1]$ if and only if $f(1)=0$.

  [If you use any theorems about uniform convergence, you should prove these.]

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• Paper 4, Section II, G

  #Analysis II

  2015

  Consider the space $\ell^{\infty}$ of bounded real sequences $x=\left(x_{i}\right)_{i=1}^{\infty}$ with the norm $\|x\|_{\infty}=\sup _{i}\left|x_{i}\right|$. Show that for every bounded sequence $x^{(n)}$ in $\ell^{\infty}$ there is a subsequence $x^{\left(n_{j}\right)}$ which converges in every coordinate, i.e. the sequence $\left(x_{i}^{\left(n_{j}\right)}\right)_{j=1}^{\infty}$ of real numbers converges for each $i$. Does every bounded sequence in $\ell^{\infty}$ have a convergent subsequence? Justify your answer.

  Let $\ell^{1} \subset \ell^{\infty}$ be the subspace of real sequences $x=\left(x_{i}\right)_{i=1}^{\infty}$ such that $\sum_{i=1}^{\infty}\left|x_{i}\right|$ converges. Is $\ell^{1}$ complete in the norm $\|\cdot\|_{\infty}$ (restricted from $\ell^{\infty}$ to $\left.\ell^{1}\right)$ ? Justify your answer.

  Suppose that $\left(x_{i}\right)$ is a real sequence such that, for every $\left(y_{i}\right) \in \ell^{\infty}$, the series $\sum_{i=1}^{\infty} x_{i} y_{i}$ converges. Show that $\left(x_{i}\right) \in \ell^{1} .$

  Suppose now that $\left(x_{i}\right)$ is a real sequence such that, for every $\left(y_{i}\right) \in \ell^{1}$, the series $\sum_{i=1}^{\infty} x_{i} y_{i}$ converges. Show that $\left(x_{i}\right) \in \ell^{\infty} .$

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• Paper 3, Section II, G

  #Analysis II

  2015

  Define what it means for a function $f: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ to be differentiable at $x \in \mathbb{R}^{n}$ with derivative $D f(x)$.

  State and prove the chain rule for the derivative of $g \circ f$, where $g: \mathbb{R}^{m} \rightarrow \mathbb{R}^{p}$ is a differentiable function.

  Now let $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ be a differentiable function and let $g(x)=f(x, c-x)$ where $c$ is a constant. Show that $g$ is differentiable and find its derivative in terms of the partial derivatives of $f$. Show that if $D_{1} f(x, y)=D_{2} f(x, y)$ holds everywhere in $\mathbb{R}^{2}$, then $f(x, y)=h(x+y)$ for some differentiable function $h .$

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• Paper 2, Section II, G

  #Analysis II

  2015

  Let $E, F$ be normed spaces with norms $\|\cdot\|_{E},\|\cdot\|_{F}$. Show that for a map $f: E \rightarrow F$ and $a \in E$, the following two statements are equivalent:

  (i) For every given $\varepsilon>0$ there exists $\delta>0$ such that $\|f(x)-f(a)\|_{F}<\varepsilon$ whenever $\|x-a\|_{E}<\delta$

  (ii) $f\left(x_{n}\right) \rightarrow f(a)$ for each sequence $x_{n} \rightarrow a$.

  We say that $f$ is continuous at $a$ if (i), or equivalently (ii), holds.

  Let now $\left(E,\|\cdot\|_{E}\right)$ be a normed space. Let $A \subset E$ be a non-empty closed subset and define $d(x, A)=\inf \left\{\|x-a\|_{E}: a \in A\right\}$. Show that

  $$|d(x, A)-d(y, A)| \leqslant\|x-y\|_{E} \text { for all } x, y \in E .$$

  In the case when $E=\mathbb{R}^{n}$ with the standard Euclidean norm, show that there exists $a \in A$ such that $d(x, A)=\|x-a\|$.

  Let $A, B$ be two disjoint closed sets in $\mathbb{R}^{n}$. Must there exist disjoint open sets $U, V$ such that $A \subset U$ and $B \subset V$ ? Must there exist $a \in A$ and $b \in B$ such that $d(a, b) \leqslant d(x, y)$ for all $x \in A$ and $y \in B$ ? For each answer, give a proof or counterexample as appropriate.

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• Paper 4, Section I, G

  #Complex Analysis

  2015

  Let $f$ be a continuous function defined on a connected open set $D \subset \mathbb{C}$. Prove carefully that the following statements are equivalent.

  (i) There exists a holomorphic function $F$ on $D$ such that $F^{\prime}(z)=f(z)$.

  (ii) $\int_{\gamma} f(z) d z=0$ holds for every closed curve $\gamma$ in $D$.

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• Paper 2, Section II, B

  #Complex Analysis or Complex Methods

  2015

  (i) A function $f(z)$ has a pole of order $m$ at $z=z_{0}$. Derive a general expression for the residue of $f(z)$ at $z=z_{0}$ involving $f$ and its derivatives.

  (ii) Using contour integration along a contour in the upper half-plane, determine the value of the integral

  $$I=\int_{0}^{\infty} \frac{(\ln x)^{2}}{\left(1+x^{2}\right)^{2}} \mathrm{~d} x$$

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• Paper 1, Section II, B

  #Complex Analysis or Complex Methods

  2015

  (i) Show that transformations of the complex plane of the form

  $$\zeta=\frac{a z+b}{c z+d}$$

  always map circles and lines to circles and lines, where $a, b, c$ and $d$ are complex numbers such that $a d-b c \neq 0$.

  (ii) Show that the transformation

  $$\zeta=\frac{z-\alpha}{\bar{\alpha} z-1}, \quad|\alpha|<1$$

  maps the unit disk centered at $z=0$ onto itself.

  (iii) Deduce a conformal transformation that maps the non-concentric annular domain $\Omega=\{|z|<1,|z-c|>c\}, 0<c<1 / 2$, to a concentric annular domain.

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• Paper 3, Section II, G

  #Complex Analysis

  2015

  State the argument principle.

  Let $U \subset \mathbb{C}$ be an open set and $f: U \rightarrow \mathbb{C}$ a holomorphic injective function. Show that $f^{\prime}(z) \neq 0$ for each $z$ in $U$ and that $f(U)$ is open.

  Stating clearly any theorems that you require, show that for each $a \in U$ and a sufficiently small $r>0$,

  $$g(w)=\frac{1}{2 \pi i} \int_{|z-a|=r} \frac{z f^{\prime}(z)}{f(z)-w} d z$$

  defines a holomorphic function on some open disc $D$ about $f(a)$.

  Show that $g$ is the inverse for the restriction of $f$ to $g(D)$.

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• Paper 1, Section I, B

  #Complex Analysis or Complex Methods

  2015

  Consider the analytic (holomorphic) functions $f$ and $g$ on a nonempty domain $\Omega$ where $g$ is nowhere zero. Prove that if $|f(z)|=|g(z)|$ for all $z$ in $\Omega$ then there exists a real constant $\alpha$ such that $f(z)=e^{i \alpha} g(z)$ for all $z$ in $\Omega$.

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• Paper 3, Section I, B

  #Complex Methods

  2015

  Find the Fourier transform of the function

  $$f(x)=\frac{1}{1+x^{2}}, \quad x \in \mathbb{R}$$

  using an appropriate contour integration. Hence find the Fourier transform of its derivative, $f^{\prime}(x)$, and evaluate the integral

  $$I=\int_{-\infty}^{\infty} \frac{4 x^{2}}{\left(1+x^{2}\right)^{4}} d x$$

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• Paper 4, Section II, B

  #Complex Methods

  2015

  (i) State and prove the convolution theorem for Laplace transforms of two realvalued functions.

  (ii) Let the function $f(t), t \geqslant 0$, be equal to 1 for $0 \leqslant t \leqslant a$ and zero otherwise, where $a$ is a positive parameter. Calculate the Laplace transform of $f$. Hence deduce the Laplace transform of the convolution $g=f * f$. Invert this Laplace transform to obtain an explicit expression for $g(t)$.

  [Hint: You may use the notation $\left.(t-a)_{+}=H(t-a) \cdot(t-a) .\right]$

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• Paper 2, Section I, A

  #Electromagnetism

  2015

  In a constant electric field $\mathbf{E}=(E, 0,0)$ a particle of rest mass $m$ and charge $q>0$ has position $\mathbf{x}$ and velocity $\dot{\mathbf{x}}$. At time $t=0$, the particle is at rest at the origin. Including relativistic effects, calculate $\dot{\mathbf{x}}(t)$.

  Sketch a graph of $|\dot{\mathbf{x}}(t)|$ versus $t$, commenting on the $t \rightarrow \infty$ limit.

  Calculate $|\mathbf{x}(t)|$ as an explicit function of $t$ and find the non-relativistic limit at small times $t$.

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• Paper 4, Section I, A

  #Electromagnetism

  2015

  From Maxwell's equations, derive the Biot-Savart law

  $$\mathbf{B}(\mathbf{r})=\frac{\mu_{0}}{4 \pi} \int_{V} \frac{\mathbf{J}\left(\mathbf{r}^{\prime}\right) \times\left(\mathbf{r}-\mathbf{r}^{\prime}\right)}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|^{3}} d^{3} \mathbf{r}^{\prime}$$

  giving the magnetic field $\mathbf{B}(\mathbf{r})$ produced by a steady current density $\mathbf{J}(\mathbf{r})$ that vanishes outside a bounded region $V$.

  [You may assume that you can choose a gauge such that the divergence of the magnetic vector potential is zero.]

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• Paper 1, Section II, A

  #Electromagnetism

  2015

  (i) Write down the Lorentz force law for $d \mathbf{p} / d t$ due to an electric field $\mathbf{E}$ and magnetic field $\mathbf{B}$ acting on a particle of charge $q$ moving with velocity $\dot{\mathbf{x}}$.

  (ii) Write down Maxwell's equations in terms of $c$ (the speed of light in a vacuum), in the absence of charges and currents.

  (iii) Show that they can be manipulated into a wave equation for each component of $\mathbf{E}$.

  (iv) Show that Maxwell's equations admit solutions of the form

  $$\mathbf{E}(\mathbf{x}, t)=\operatorname{Re}\left(\mathbf{E}_{0} e^{i(\omega t-\mathbf{k} \cdot \mathbf{x})}\right)$$

  where $\mathbf{E}_{\mathbf{0}}$ and $\mathbf{k}$ are constant vectors and $\omega$ is a constant (all real). Derive a condition on $\mathbf{k} \cdot \mathbf{E}_{\mathbf{0}}$ and relate $\omega$ and $\mathbf{k}$.

  (v) Suppose that a perfect conductor occupies the region $z<0$ and that a plane wave with $\mathbf{k}=(0,0,-k), \mathbf{E}_{0}=\left(E_{0}, 0,0\right)$ is incident from the vacuum region $z>0$. Write down boundary conditions for the $\mathbf{E}$ and $\mathbf{B}$ fields. Show that they can be satisfied if a suitable reflected wave is present, and determine the total $\mathbf{E}$ and $\mathbf{B}$ fields in real form.

  (vi) At time $t=\pi /(4 \omega)$, a particle of charge $q$ and mass $m$ is at $(0,0, \pi /(4 k))$ moving with velocity $(c / 2,0,0)$. You may assume that the particle is far enough away from the conductor so that we can ignore its effect upon the conductor and that $q E_{0}>0$. Give a unit vector for the direction of the Lorentz force on the particle at time $t=\pi /(4 \omega)$.

  (vii) Ignoring relativistic effects, find the magnitude of the particle's rate of change of velocity in terms of $E_{0}, q$ and $m$ at time $t=\pi /(4 \omega)$. Why is this answer inaccurate?

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• Paper 3, Section II, A

  #Electromagnetism

  2015

  A charge density $\rho=\lambda / r$ fills the region of 3-dimensional space $a<r<b$, where $r$ is the radial distance from the origin and $\lambda$ is a constant. Compute the electric field in all regions of space in terms of $Q$, the total charge of the region. Sketch a graph of the magnitude of the electric field versus $r$ (assuming that $Q>0$ ).

  Now let $\Delta=b-a \rightarrow 0$. Derive the surface charge density $\sigma$ in terms of $\Delta, a$ and $\lambda$ and explain how a finite surface charge density may be obtained in this limit. Sketch the magnitude of the electric field versus $r$ in this limit. Comment on any discontinuities, checking a standard result involving $\sigma$ for this particular case.

  A second shell of equal and opposite total charge is centred on the origin and has a radius $c<a$. Sketch the electric potential of this system, assuming that it tends to 0 as $r \rightarrow \infty$.

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• Paper 2, Section II, A

  #Electromagnetism

  2015

  Consider the magnetic field

  $$\mathbf{B}=b[\mathbf{r}+(k \hat{\mathbf{z}}+l \hat{\mathbf{y}}) \hat{\mathbf{z}} \cdot \mathbf{r}+p \hat{\mathbf{x}}(\hat{\mathbf{y}} \cdot \mathbf{r})+n \hat{\mathbf{z}}(\hat{\mathbf{x}} \cdot \mathbf{r})],$$

  where $b \neq 0, \mathbf{r}=(x, y, z)$ and $\hat{\mathbf{x}}, \hat{\mathbf{y}}, \hat{\mathbf{z}}$ are unit vectors in the $x, y$ and $z$ directions, respectively. Imposing that this satisfies the expected equations for a static magnetic field in a vacuum, find $k, l, n$ and $p$.

  A circular wire loop of radius $a$, mass $m$ and resistance $R$ lies in the $(x, y)$ plane with its centre on the $z$-axis at $z$ and a magnetic field as given above. Calculate the magnetic flux through the loop arising from this magnetic field and also the force acting on the loop when a current $I$ is flowing around the loop in a clockwise direction about the $z$-axis.

  At $t=0$, the centre of the loop is at the origin, travelling with velocity $(0,0, v(t=0))$, where $v(0)>0$. Ignoring gravity and relativistic effects, and assuming that $I$ is only the induced current, find the time taken for the speed to halve in terms of $a, b, R$ and $m$. By what factor does the rate of heat generation change in this time?

  Where is the loop as $t \rightarrow \infty$ as a function of $a, b, R, v(0) ?$

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• Paper 1, Section I, B

  #Fluid Dynamics

  2015

  Consider a spherical bubble of radius $a$ in an inviscid fluid in the absence of gravity. The flow at infinity is at rest and the bubble undergoes translation with velocity $\mathbf{U}=U(t) \hat{\mathbf{x}}$. We assume that the flow is irrotational and derives from a potential given in spherical coordinates by

  $$\phi(r, \theta)=U(t) \frac{a^{3}}{2 r^{2}} \cos \theta,$$

  where $\theta$ is measured with respect to $\hat{\mathbf{x}}$. Compute the force, $\mathbf{F}$, acting on the bubble. Show that the formula for $\mathbf{F}$ can be interpreted as the acceleration force of a fraction $\alpha<1$ of the fluid displaced by the bubble, and determine the value of $\alpha$.

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• Paper 2, Section I, B

  #Fluid Dynamics

  2015

  Consider the two-dimensional velocity field $\mathbf{u}=(u, v)$ with

  $$u(x, y)=x^{2}-y^{2}, \quad v(x, y)=-2 x y$$

  (i) Show that the flow is incompressible and irrotational.

  (ii) Derive the velocity potential, $\phi$, and the streamfunction, $\psi$.

  (iii) Plot all streamlines passing through the origin.

  (iv) Show that the complex function $w=\phi+i \psi$ (where $i^{2}=-1$ ) can be written solely as a function of the complex coordinate $z=x+i y$ and determine that function.

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• Paper 1, Section II, B

  #Fluid Dynamics

  2015

  A fluid layer of depth $h_{1}$ and dynamic viscosity $\mu_{1}$ is located underneath a fluid layer of depth $h_{2}$ and dynamic viscosity $\mu_{2}$. The total fluid system of depth $h=h_{1}+h_{2}$ is positioned between a stationary rigid plate at $y=0$ and a rigid plate at $y=h$ moving with speed $\mathbf{U}=U \hat{\mathbf{x}}$, where $U$ is constant. Ignore the effects of gravity.

  (i) Using dimensional analysis only, and the fact that the stress should be linear in $U$, derive the expected form of the shear stress acted by the fluid on the plate at $y=0$ as a function of $U, h_{1}, h_{2}, \mu_{1}$ and $\mu_{2}$.

  (ii) Solve for the unidirectional velocity profile between the two plates. State clearly all boundary conditions you are using to solve this problem.

  (iii) Compute the exact value of the shear stress acted by the fluid on the plate at $y=0$. Compare with the results in (i).

  (iv) What is the condition on the viscosity of the bottom layer, $\mu_{1}$, for the stress in (iii) to be smaller than it would be if the fluid had constant viscosity $\mu_{2}$ in both layers?

  (v) Show that the stress acting on the plate at $y=h$ is equal and opposite to the stress on the plate at $y=0$ and justify this result physically.

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• Paper 4, Section II, B

  #Fluid Dynamics

  2015

  Consider a steady inviscid, incompressible fluid of constant density $\rho$ in the absence of external body forces. A cylindrical jet of area $A$ and speed $U$ impinges fully on a stationary sphere of radius $R$ with $A<\pi R^{2}$. The flow is assumed to remain axisymmetric and be deflected into a conical sheet of vertex angle $\alpha>0$.

  (i) Show that the speed of the fluid in the conical sheet is constant.

  (ii) Use conservation of mass to show that the width $d(r)$ of the fluid sheet at a distance $r \gg R$ from point of impact is given by

  $$d(r)=\frac{A}{2 \pi r \sin \alpha}$$

  (iii) Use Euler's equation to derive the momentum integral equation

  $$\iint_{S}\left(p n_{i}+\rho n_{j} u_{j} u_{i}\right) d S=0$$

  for a closed surface $S$ with normal $\mathbf{n}$ where $u_{m}$ is the $m$ th component of the velocity field in cartesian coordinates and $p$ is the pressure.

  (iv) Use the result from (iii) to calculate the net force on the sphere.

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• Paper 3, Section II, B

  #Fluid Dynamics

  2015

  A source of sound induces a travelling wave of pressure above the free surface of a fluid located in the $z<0$ domain as

  $$p=p_{a t m}+p_{0} \cos (k x-\omega t),$$

  with $p_{0} \ll p_{a t m}$. Here $k$ and $\omega$ are fixed real numbers. We assume that the flow induced in the fluid is irrotational.

  (i) State the linearized equation of motion for the fluid and the free surface, $z=h(x, t)$, with all boundary conditions.

  (ii) Solve for the velocity potential, $\phi(x, z, t)$, and the height of the free surface, $h(x, t)$. Verify that your solutions are dimensionally correct.

  (iii) Interpret physically the behaviour of the solution when $\omega^{2}=g k$.

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• Paper 1, Section I, F

  #Geometry

  2015

  (i) Give a model for the hyperbolic plane. In this choice of model, describe hyperbolic lines.

  Show that if $\ell_{1}, \ell_{2}$ are two hyperbolic lines and $p_{1} \in \ell_{1}, p_{2} \in \ell_{2}$ are points, then there exists an isometry $g$ of the hyperbolic plane such that $g\left(\ell_{1}\right)=\ell_{2}$ and $g\left(p_{1}\right)=p_{2}$.

  (ii) Let $T$ be a triangle in the hyperbolic plane with angles $30^{\circ}, 30^{\circ}$ and $45^{\circ}$. What is the area of $T$ ?

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• Paper 3, Section $I$, F

  #Geometry

  2015

  State the sine rule for spherical triangles.

  Let $\Delta$ be a spherical triangle with vertices $A, B$, and $C$, with angles $\alpha, \beta$ and $\gamma$ at the respective vertices. Let $a, b$, and $c$ be the lengths of the edges $B C, A C$ and $A B$ respectively. Show that $b=c$ if and only if $\beta=\gamma$. [You may use the cosine rule for spherical triangles.] Show that this holds if and only if there exists a reflection $M$ such that $M(A)=A, M(B)=C$ and $M(C)=B$.

  Are there equilateral triangles on the sphere? Justify your answer.

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• Paper 3, Section II, F

  #Geometry

  2015

  Let $T: \mathbb{C}_{\infty} \rightarrow \mathbb{C}_{\infty}$ be a Möbius transformation on the Riemann sphere $\mathbb{C}_{\infty}$.

  (i) Show that $T$ has either one or two fixed points.

  (ii) Show that if $T$ is a Möbius transformation corresponding to (under stereographic projection) a rotation of $S^{2}$ through some fixed non-zero angle, then $T$ has two fixed points, $z_{1}, z_{2}$, with $z_{2}=-1 / \bar{z}_{1}$.

  (iii) Suppose $T$ has two fixed points $z_{1}, z_{2}$ with $z_{2}=-1 / \bar{z}_{1}$. Show that either $T$ corresponds to a rotation as in (ii), or one of the fixed points, say $z_{1}$, is attractive, i.e. $T^{n} z \rightarrow z_{1}$ as $n \rightarrow \infty$ for any $z \neq z_{2}$.

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• Paper 2, Section II, F

  #Geometry

  2015

  (a) For each of the following subsets of $\mathbb{R}^{3}$, explain briefly why it is a smooth embedded surface or why it is not.

  $$\begin{aligned} S_{1} &=\{(x, y, z) \mid x=y, z=3\} \cup\{(2,3,0)\} \\ S_{2} &=\left\{(x, y, z) \mid x^{2}+y^{2}-z^{2}=1\right\} \\ S_{3} &=\left\{(x, y, z) \mid x^{2}+y^{2}-z^{2}=0\right\} \end{aligned}$$

  (b) Let $f: U=\{(u, v) \mid v>0\} \rightarrow \mathbb{R}^{3}$ be given by

  $$f(u, v)=\left(u^{2}, u v, v\right),$$

  and let $S=f(U) \subseteq \mathbb{R}^{3}$. You may assume that $S$ is a smooth embedded surface.

  Find the first fundamental form of this surface.

  Find the second fundamental form of this surface.

  Compute the Gaussian curvature of this surface.

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• Paper 4, Section II, F

  #Geometry

  2015

  Let $\alpha(s)=(f(s), g(s))$ be a curve in $\mathbb{R}^{2}$ parameterized by arc length, and consider the surface of revolution $S$ in $\mathbb{R}^{3}$ defined by the parameterization

  $$\sigma(u, v)=(f(u) \cos v, f(u) \sin v, g(u))$$

  In what follows, you may use that a curve $\sigma \circ \gamma$ in $S$, with $\gamma(t)=(u(t), v(t))$, is a geodesic if and only if

  $$\ddot{u}=f(u) \frac{d f}{d u} \dot{v}^{2}, \quad \frac{d}{d t}\left(f(u)^{2} \dot{v}\right)=0$$

  (i) Write down the first fundamental form for $S$, and use this to write down a formula which is equivalent to $\sigma \circ \gamma$ being a unit speed curve.

  (ii) Show that for a given $u_{0}$, the circle on $S$ determined by $u=u_{0}$ is a geodesic if and only if $\frac{d f}{d u}\left(u_{0}\right)=0$.

  (iii) Let $\gamma(t)=(u(t), v(t))$ be a curve in $\mathbb{R}^{2}$ such that $\sigma \circ \gamma$ parameterizes a unit speed curve that is a geodesic in $S$. For a given time $t_{0}$, let $\theta\left(t_{0}\right)$ denote the angle between the curve $\sigma \circ \gamma$ and the circle on $S$ determined by $u=u\left(t_{0}\right)$. Derive Clairault's relation that

  $$f(u(t)) \cos (\theta(t))$$

  is independent of $t$.

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• Paper 3, Section I, F

  #Groups, Rings and Modules

  2015

  State two equivalent conditions for a commutative ring to be Noetherian, and prove they are equivalent. Give an example of a ring which is not Noetherian, and explain why it is not Noetherian.

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• Paper 4, Section I, $2 F$

  #Groups, Rings and Modules

  2015

  Let $R$ be a commutative ring. Define what it means for an ideal $I \subseteq R$ to be prime. Show that $I \subseteq R$ is prime if and only if $R / I$ is an integral domain.

  Give an example of an integral domain $R$ and an ideal $I \subset R, I \neq R$, such that $R / I$ is not an integral domain.

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• Paper 2, Section $I$, $2 F$

  #Groups, Rings and Modules

  2015

  Give four non-isomorphic groups of order 12 , and explain why they are not isomorphic.

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• Paper 1, Section II, F

  #Groups, Rings and Modules

  2015

  (i) Give the definition of a $p$-Sylow subgroup of a group.

  (ii) Let $G$ be a group of order $2835=3^{4} \cdot 5 \cdot 7$. Show that there are at most two possibilities for the number of 3-Sylow subgroups, and give the possible numbers of 3-Sylow subgroups.

  (iii) Continuing with a group $G$ of order 2835 , show that $G$ is not simple.

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• Paper 4, Section II, F

  #Groups, Rings and Modules

  2015

  Find $a \in \mathbb{Z}_{7}$ such that $\mathbb{Z}_{7}[x] /\left(x^{3}+a\right)$ is a field $F$. Show that for your choice of $a$, every element of $\mathbb{Z}_{7}$ has a cube root in the field $F$.

  Show that if $F$ is a finite field, then the multiplicative group $F^{\times}=F \backslash\{0\}$ is cyclic.

  Show that $F=\mathbb{Z}_{2}[x] /\left(x^{3}+x+1\right)$ is a field. How many elements does $F$ have? Find a generator for $F^{\times}$.

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• Paper 3, Section II, F

  #Groups, Rings and Modules

  2015

  Can a group of order 55 have 20 elements of order 11? If so, give an example. If not, give a proof, including the proof of any statements you need.

  Let $G$ be a group of order $p q$, with $p$ and $q$ primes, $p>q$. Suppose furthermore that $q$ does not divide $p-1$. Show that $G$ is cyclic.

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• Paper 2, Section II, F

  #Groups, Rings and Modules

  2015

  (a) Consider the homomorphism $f: \mathbb{Z}^{3} \rightarrow \mathbb{Z}^{4}$ given by

  $$f(a, b, c)=(a+2 b+8 c, 2 a-2 b+4 c,-2 b+12 c, 2 a-4 b+4 c)$$

  Describe the image of this homomorphism as an abstract abelian group. Describe the quotient of $\mathbb{Z}^{4}$ by the image of this homomorphism as an abstract abelian group.

  (b) Give the definition of a Euclidean domain.

  Fix a prime $p$ and consider the subring $R$ of the rational numbers $\mathbb{Q}$ defined by

  $$R=\{q / r \mid \operatorname{gcd}(p, r)=1\}$$

  where 'gcd' stands for the greatest common divisor. Show that $R$ is a Euclidean domain.

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• Paper 4, Section I, E

  #Linear Algebra

  2015

  Define the dual space $V^{*}$ of a vector space $V$. Given a basis $\left\{x_{1}, \ldots, x_{n}\right\}$ of $V$ define its dual and show it is a basis of $V^{*}$.

  Let $V$ be a 3-dimensional vector space over $\mathbb{R}$ and let $\left\{\zeta_{1}, \zeta_{2}, \zeta_{3}\right\}$ be the basis of $V^{*}$ dual to the basis $\left\{x_{1}, x_{2}, x_{3}\right\}$ for $V$. Determine, in terms of the $\zeta_{i}$, the bases dual to each of the following: (a) $\left\{x_{1}+x_{2}, x_{2}+x_{3}, x_{3}\right\}$, (b) $\left\{x_{1}+x_{2}, x_{2}+x_{3}, x_{3}+x_{1}\right\}$.

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• Paper 2, Section I, $1 \mathrm{E}$

  #Linear Algebra

  2015

  Let $q$ denote a quadratic form on a real vector space $V$. Define the rank and signature of $q$.

  Find the rank and signature of the following quadratic forms. (a) $q(x, y, z)=x^{2}+y^{2}+z^{2}-2 x z-2 y z$. (b) $q(x, y, z)=x y-x z$.

  (c) $q(x, y, z)=x y-2 z^{2}$.

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• Paper 1, Section I, E

  #Linear Algebra

  2015

  Let $U$ and $V$ be finite dimensional vector spaces and $\alpha: U \rightarrow V$ a linear map. Suppose $W$ is a subspace of $U$. Prove that

  $$r(\alpha) \geqslant r\left(\left.\alpha\right|_{W}\right) \geqslant r(\alpha)-\operatorname{dim}(U)+\operatorname{dim}(W)$$

  where $r(\alpha)$ denotes the rank of $\alpha$ and $\left.\alpha\right|_{W}$ denotes the restriction of $\alpha$ to $W$. Give examples showing that each inequality can be both a strict inequality and an equality.

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• Paper 1, Section II, E

  #Linear Algebra

  2015

  Determine the characteristic polynomial of the matrix

  $$M=\left(\begin{array}{cccc} x & 1 & 1 & 0 \\ 1-x & 0 & -1 & 0 \\ 2 & 2 x & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right)$$

  For which values of $x \in \mathbb{C}$ is $M$ invertible? When $M$ is not invertible determine (i) the Jordan normal form $J$ of $M$, (ii) the minimal polynomial of $M$.

  Find a basis of $\mathbb{C}^{4}$ such that $J$ is the matrix representing the endomorphism $M: \mathbb{C}^{4} \rightarrow \mathbb{C}^{4}$ in this basis. Give a change of basis matrix $P$ such that $P^{-1} M P=J$.

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• Paper 4, Section II, E

  #Linear Algebra

  2015

  Suppose $U$ and $W$ are subspaces of a vector space $V$. Explain what is meant by $U \cap W$ and $U+W$ and show that both of these are subspaces of $V$.

  Show that if $U$ and $W$ are subspaces of a finite dimensional space $V$ then

  $$\operatorname{dim} U+\operatorname{dim} W=\operatorname{dim}(U \cap W)+\operatorname{dim}(U+W)$$

  Determine the dimension of the subspace $W$ of $\mathbb{R}^{5}$ spanned by the vectors

  $$\left(\begin{array}{c} 1 \\ 3 \\ 3 \\ -1 \\ 1 \end{array}\right),\left(\begin{array}{l} 4 \\ 1 \\ 3 \\ 2 \\ 1 \end{array}\right),\left(\begin{array}{l} 3 \\ 2 \\ 1 \\ 2 \\ 3 \end{array}\right),\left(\begin{array}{c} 2 \\ 2 \\ 5 \\ -1 \\ -1 \end{array}\right)$$

  Write down a $5 \times 5$ matrix which defines a linear map $\mathbb{R}^{5} \rightarrow \mathbb{R}^{5}$ with $(1,1,1,1,1)^{T}$ in the kernel and with image $W$.

  What is the dimension of the space spanned by all linear maps $\mathbb{R}^{5} \rightarrow \mathbb{R}^{5}$

  (i) with $(1,1,1,1,1)^{T}$ in the kernel and with image contained in $W$,

  (ii) with $(1,1,1,1,1)^{T}$ in the kernel or with image contained in $W$ ?

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• Paper 3, Section II, E

  #Linear Algebra

  2015

  Let $A_{1}, A_{2}, \ldots, A_{k}$ be $n \times n$ matrices over a field $\mathbb{F}$. We say $A_{1}, A_{2}, \ldots, A_{k}$ are simultaneously diagonalisable if there exists an invertible matrix $P$ such that $P^{-1} A_{i} P$ is diagonal for all $1 \leqslant i \leqslant k$. We say the matrices are commuting if $A_{i} A_{j}=A_{j} A_{i}$ for all $i, j$.

  (i) Suppose $A_{1}, A_{2}, \ldots, A_{k}$ are simultaneously diagonalisable. Prove that they are commuting.

  (ii) Define an eigenspace of a matrix. Suppose $B_{1}, B_{2}, \ldots, B_{k}$ are commuting $n \times n$ matrices over a field $\mathbb{F}$. Let $E$ denote an eigenspace of $B_{1}$. Prove that $B_{i}(E) \leqslant E$ for all $i$.

  (iii) Suppose $B_{1}, B_{2}, \ldots, B_{k}$ are commuting diagonalisable matrices. Prove that they are simultaneously diagonalisable.

  (iv) Are the $2 \times 2$ diagonalisable matrices over $\mathbb{C}$ simultaneously diagonalisable? Explain your answer.

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• Paper 2, Section II, E

  #Linear Algebra

  2015

  (i) Suppose $A$ is a matrix that does not have $-1$ as an eigenvalue. Show that $A+I$ is non-singular. Further, show that $A$ commutes with $(A+I)^{-1}$.

  (ii) A matrix $A$ is called skew-symmetric if $A^{T}=-A$. Show that a real skewsymmetric matrix does not have $-1$ as an eigenvalue.

  (iii) Suppose $A$ is a real skew-symmetric matrix. Show that $U=(I-A)(I+A)^{-1}$ is orthogonal with determinant 1 .

  (iv) Verify that every orthogonal matrix $U$ with determinant 1 which does not have $-1$ as an eigenvalue can be expressed as $(I-A)(I+A)^{-1}$ where $A$ is a real skew-symmetric matrix.

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• Paper 4, Section I, H

  #Markov Chains

  2015

  Let $X_{0}, X_{1}, X_{2}, \ldots$ be independent identically distributed random variables with $\mathbb{P}\left(X_{i}=1\right)=1-\mathbb{P}\left(X_{i}=0\right)=p, 0<p<1$. Let $Z_{n}=X_{n-1}+c X_{n}, n=1,2, \ldots$, where $c$ is a constant. For each of the following cases, determine whether or not $\left(Z_{n}: n \geqslant 1\right)$ is a Markov chain: (a) $c=0$; (b) $c=1$; (c) $c=2$.

  In each case, if $\left(Z_{n}: n \geqslant 1\right)$ is a Markov chain, explain why, and give its state space and transition matrix; if it is not a Markov chain, give an example to demonstrate that it is not.

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• Paper 3, Section I, H

  #Markov Chains

  2015

  Define what is meant by a communicating class and a closed class in a Markov chain.

  A Markov chain $\left(X_{n}: n \geqslant 0\right)$ with state space $\{1,2,3,4\}$ has transition matrix

  $$P=\left(\begin{array}{cccc} \frac{1}{2} & 0 & \frac{1}{2} & 0 \\ 0 & \frac{1}{2} & 0 & \frac{1}{2} \\ \frac{1}{2} & 0 & \frac{1}{2} & 0 \\ \frac{1}{2} & 0 & 0 & \frac{1}{2} \end{array}\right)$$

  Write down the communicating classes for this Markov chain and state whether or not each class is closed.

  If $X_{0}=2$, let $N$ be the smallest $n$ such that $X_{n} \neq 2$. Find $\mathbb{P}(N=n)$ for $n=1,2, \ldots$ and $\mathbb{E}(N)$. Describe the evolution of the chain if $X_{0}=2$.

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• Paper 2, Section II, H

  #Markov Chains

  2015

  (a) What does it mean for a transition matrix $P$ and a distribution $\lambda$ to be in detailed balance? Show that if $P$ and $\lambda$ are in detailed balance then $\lambda=\lambda P$.

  (b) A mathematician owns $r$ bicycles, which she sometimes uses for her journey from the station to College in the morning and for the return journey in the evening. If it is fine weather when she starts a journey, and if there is a bicycle available at the current location, then she cycles; otherwise she takes the bus. Assume that with probability $p$, $0<p<1$, it is fine when she starts a journey, independently of all other journeys. Let $X_{n}$ denote the number of bicycles at the current location, just before the mathematician starts the $n$th journey.

  (i) Show that $\left(X_{n} ; n \geqslant 0\right)$ is a Markov chain and write down its transition matrix.

  (ii) Find the invariant distribution of the Markov chain.

  (iii) Show that the Markov chain satisfies the necessary conditions for the convergence theorem for Markov chains and find the limiting probability that the mathematician's $n$th journey is by bicycle.

  [Results from the course may be used without proof provided that they are clearly stated.]

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• Paper 1, Section II, H

  #Markov Chains

  2015

  Consider a particle moving between the vertices of the graph below, taking steps along the edges. Let $X_{n}$ be the position of the particle at time $n$. At time $n+1$ the particle moves to one of the vertices adjoining $X_{n}$, with each of the adjoining vertices being equally likely, independently of previous moves. Explain briefly why $\left(X_{n} ; n \geqslant 0\right)$ is a Markov chain on the vertices. Is this chain irreducible? Find an invariant distribution for this chain.

  

  Suppose that the particle starts at $B$. By adapting the transition matrix, or otherwise, find the probability that the particle hits vertex $A$ before vertex $F$.

  Find the expected first passage time from $B$ to $F$ given no intermediate visit to $A$.

  [Results from the course may be used without proof provided that they are clearly stated.]

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• Paper 4, Section I, 5C

  #Methods

  2015

  (a) The convolution $f * g$ of two functions $f, g: \mathbb{R} \rightarrow \mathbb{C}$ is related to their Fourier transforms $\tilde{f}, \tilde{g}$ by

  $$\frac{1}{2 \pi} \int_{-\infty}^{\infty} \tilde{f}(k) \tilde{g}(k) e^{i k x} d k=\int_{-\infty}^{\infty} f(u) g(x-u) d u$$

  Derive Parseval's theorem for Fourier transforms from this relation.

  (b) Let $a>0$ and

  $$f(x)= \begin{cases}\cos x & \text { for } x \in[-a, a] \\ 0 & \text { elsewhere }\end{cases}$$

  (i) Calculate the Fourier transform $\tilde{f}(k)$ of $f(x)$.

  (ii) Determine how the behaviour of $\tilde{f}(k)$ in the limit $|k| \rightarrow \infty$ depends on the value of $a$. Briefly interpret the result.

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• Paper 2, Section I, C

  #Methods

  2015

  (i) Write down the trigonometric form for the Fourier series and its coefficients for a function $f:[-L, L) \rightarrow \mathbb{R}$ extended to a $2 L$-periodic function on $\mathbb{R}$.

  (ii) Calculate the Fourier series on $[-\pi, \pi)$ of the function $f(x)=\sin (\lambda x)$ where $\lambda$ is a real constant. Take the limit $\lambda \rightarrow k$ with $k \in \mathbb{Z}$ in the coefficients of this series and briefly interpret the resulting expression.

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• Paper 3, Section I, $7 \mathrm{C}$

  #Methods

  2015

  (a) From the defining property of the $\delta$ function,

  $$\int_{-\infty}^{\infty} \delta(x) f(x) d x=f(0)$$

  for any function $f$, prove that

  (i) $\delta(-x)=\delta(x)$

  (ii) $\delta(a x)=|a|^{-1} \delta(x)$ for $a \in \mathbb{R}, a \neq 0$,

  (iii) If $g: \mathbb{R} \rightarrow \mathbb{R}, x \mapsto g(x)$ is smooth and has isolated zeros $x_{i}$ where the derivative $g^{\prime}\left(x_{i}\right) \neq 0$, then

  $$\delta[g(x)]=\sum_{i} \frac{\delta\left(x-x_{i}\right)}{\left|g^{\prime}\left(x_{i}\right)\right|}$$

  (b) Show that the function $\gamma(x)$ defined by

  $$\gamma(x)=\lim _{s \rightarrow 0} \frac{e^{x / s}}{s\left(1+e^{x / s}\right)^{2}}$$

  is the $\delta(x)$ function.

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• Paper 1, Section II, C

  #Methods

  2015

  (i) Briefly describe the Sturm-Liouville form of an eigenfunction equation for real valued functions with a linear, second-order ordinary differential operator. Briefly summarize the properties of the solutions.

  (ii) Derive the condition for self-adjointness of the differential operator in (i) in terms of the boundary conditions of solutions $y_{1}, y_{2}$ to the Sturm-Liouville equation. Give at least three types of boundary conditions for which the condition for self-adjointness is satisfied.

  (iii) Consider the inhomogeneous Sturm-Liouville equation with weighted linear term

  $$\frac{1}{w(x)} \frac{d}{d x}\left(p(x) \frac{d y}{d x}\right)-\frac{q(x)}{w(x)} y-\lambda y=f(x),$$

  on the interval $a \leqslant x \leqslant b$, where $p$ and $q$ are real functions on $[a, b]$ and $w$ is the weighting function. Let $G(x, \xi)$ be a Green's function satisfying

  $$\frac{d}{d x}\left(p(x) \frac{d G}{d x}\right)-q(x) G(x, \xi)=\delta(x-\xi)$$

  Let solutions $y$ and the Green's function $G$ satisfy the same boundary conditions of the form $\alpha y^{\prime}+\beta y=0$ at $x=a, \mu y^{\prime}+\nu y=0$ at $x=b(\alpha, \beta$ are not both zero and $\mu, \nu$ are not both zero) and likewise for $G$ for the same constants $\alpha, \beta, \mu$ and $\nu$. Show that the Sturm-Liouville equation can be written as a so-called Fredholm integral equation of the form

  $$\psi(\xi)=U(\xi)+\lambda \int_{a}^{b} K(x, \xi) \psi(x) d x$$

  where $K(x, \xi)=\sqrt{w(\xi) w(x)} G(x, \xi), \psi=\sqrt{w} y$ and $U$ depends on $K, w$ and the forcing term $f$. Write down $U$ in terms of an integral involving $f, K$ and $w$.

  (iv) Derive the Fredholm integral equation for the Sturm-Liouville equation on the interval $[0,1]$

  $$\frac{d^{2} y}{d x^{2}}-\lambda y=0,$$

  with $y(0)=y(1)=0$.

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• Paper 3, Section II, C

  #Methods

  2015

  (i) Consider the Poisson equation $\nabla^{2} \psi(\mathbf{r})=f(\mathbf{r})$ with forcing term $f$ on the infinite domain $\mathbb{R}^{3}$ with $\lim _{|\mathbf{r}| \rightarrow \infty} \psi=0$. Derive the Green's function $G\left(\mathbf{r}, \mathbf{r}^{\prime}\right)=-1 /\left(4 \pi\left|\mathbf{r}-\mathbf{r}^{\prime}\right|\right)$ for this equation using the divergence theorem. [You may assume without proof that the divergence theorem is valid for the Green's function.]

  (ii) Consider the Helmholtz equation

  $$\nabla^{2} \psi(\mathbf{r})+k^{2} \psi(\mathbf{r})=f(\mathbf{r})$$

  where $k$ is a real constant. A Green's function $g\left(\mathbf{r}, \mathbf{r}^{\prime}\right)$ for this equation can be constructed from $G\left(\mathbf{r}, \mathbf{r}^{\prime}\right)$ of (i) by assuming $g\left(\mathbf{r}, \mathbf{r}^{\prime}\right)=U(r) G\left(\mathbf{r}, \mathbf{r}^{\prime}\right)$ where $r=\left|\mathbf{r}-\mathbf{r}^{\prime}\right|$ and $U(r)$ is a regular function. Show that $\lim _{r \rightarrow 0} U(r)=1$ and that $U$ satisfies the equation

  $$\frac{d^{2} U}{d r^{2}}+k^{2} U(r)=0$$

  (iii) Take the Green's function with the specific solution $U(r)=e^{i k r}$ to Eq. ( $\ddagger$ ) and consider the Helmholtz equation $(\dagger)$ on the semi-infinite domain $z>0, x, y \in \mathbb{R}$. Use the method of images to construct a Green's function for this problem that satisfies the boundary conditions

  $$\frac{\partial g}{\partial z^{\prime}}=0 \text { on } z^{\prime}=0 \quad \text { and } \quad \lim _{|\mathbf{r}| \rightarrow \infty} g\left(\mathbf{r}, \mathbf{r}^{\prime}\right)=0$$

  (iv) A solution to the Helmholtz equation on a bounded domain can be constructed in complete analogy to that of the Poisson equation using the Green's function in Green's 3rd identity

  $$\psi(\mathbf{r})=\int_{\partial V}\left[\psi\left(\mathbf{r}^{\prime}\right) \frac{\partial g\left(\mathbf{r}, \mathbf{r}^{\prime}\right)}{\partial n^{\prime}}-g\left(\mathbf{r}, \mathbf{r}^{\prime}\right) \frac{\partial \psi\left(\mathbf{r}^{\prime}\right)}{\partial n^{\prime}}\right] d S^{\prime}+\int_{V} f\left(\mathbf{r}^{\prime}\right) g\left(\mathbf{r}, \mathbf{r}^{\prime}\right) d V^{\prime},$$

  where $V$ denotes the volume of the domain, $\partial V$ its boundary and $\partial / \partial n^{\prime}$ the outgoing normal derivative on the boundary. Now consider the homogeneous Helmholtz equation $\nabla^{2} \psi(\mathbf{r})+k^{2} \psi(\mathbf{r})=0$ on the domain $z>0, x, y \in \mathbb{R}$ with boundary conditions $\psi(\mathbf{r})=0$ at $|\mathbf{r}| \rightarrow \infty$ and

  $$\left.\frac{\partial \psi}{\partial z}\right|_{z=0}= \begin{cases}0 & \text { for } \rho>a \\ A & \text { for } \rho \leqslant a\end{cases}$$

  where $\rho=\sqrt{x^{2}+y^{2}}$ and $A$ and $a$ are real constants. Construct a solution in integral form to this equation using cylindrical coordinates $(z, \rho, \varphi)$ with $x=\rho \cos \varphi, y=\rho \sin \varphi$.

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• Paper 2, Section II, C

  #Methods

  2015

  (i) The Laplace operator in spherical coordinates is

  $$\vec{\nabla}^{2}=\frac{1}{r^{2}} \frac{\partial}{\partial r}\left(r^{2} \frac{\partial}{\partial r}\right)+\frac{1}{r^{2} \sin \theta} \frac{\partial}{\partial \theta}\left(\sin \theta \frac{\partial}{\partial \theta}\right)+\frac{1}{r^{2} \sin ^{2} \theta} \frac{\partial^{2}}{\partial \phi^{2}}$$

  Show that general, regular axisymmetric solutions $\psi(r, \theta)$ to the equation $\vec{\nabla}^{2} \psi=0$ are given by

  $$\psi(r, \theta)=\sum_{n=0}^{\infty}\left(A_{n} r^{n}+B_{n} r^{-(n+1)}\right) P_{n}(\cos \theta)$$

  where $A_{n}, B_{n}$ are constants and $P_{n}$ are the Legendre polynomials. [You may use without proof that regular solutions to Legendre's equation $-\frac{d}{d x}\left[\left(1-x^{2}\right) \frac{d}{d x} y(x)\right]=\lambda y(x)$ are given by $P_{n}(x)$ with $\lambda=n(n+1)$ and non-negative integer $n$.]

  (ii) Consider a uniformly charged wire in the form of a ring of infinitesimal width with radius $r_{0}=1$ and a constant charge per unit length $\sigma$. By Coulomb's law, the electric potential due to a point charge $q$ at a point a distance $d$ from the charge is

  $$U=\frac{q}{4 \pi \epsilon_{0} d}$$

  where $\epsilon_{0}$ is a constant. Let the $z$-axis be perpendicular to the circle and pass through the circle's centre (see figure). Show that the potential due to the charged ring at a point on the $z$-axis at location $z$ is given by

  

  $$V=\frac{\sigma}{2 \epsilon_{0} \sqrt{1+z^{2}}} .$$

  (iii) The potential $V$ generated by the charged ring of (ii) at arbitrary points (excluding points directly on the ring which can be ignored for this question) is determined by Laplace's equation $\vec{\nabla}^{2} V=0$. Calculate this potential with the boundary condition $\lim _{r \rightarrow \infty} V=0$, where $r=\sqrt{x^{2}+y^{2}+z^{2}}$. [You may use without proof that

  $$\frac{1}{\sqrt{1+x^{2}}}=\sum_{m=0}^{\infty} x^{2 m}(-1)^{m} \frac{(2 m) !}{2^{2 m}(m !)^{2}}$$

  for $|x|<1$. Furthermore, the Legendre polynomials are normalized such that $\left.P_{n}(1)=1 .\right]$

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• Paper 4, Section II, 17C

  #Methods

  2015

  Describe the method of characteristics to construct solutions for 1st-order, homogeneous, linear partial differential equations

  $$\alpha(x, y) \frac{\partial u}{\partial x}+\beta(x, y) \frac{\partial u}{\partial y}=0$$

  with initial data prescribed on a curve $x_{0}(\sigma), y_{0}(\sigma): u\left(x_{0}(\sigma), y_{0}(\sigma)\right)=h(\sigma)$.

  Consider the partial differential equation (here the two independent variables are time $t$ and spatial direction $x$ )

  $$\frac{\partial u}{\partial t}+u \frac{\partial u}{\partial x}=0$$

  with initial data $u(t=0, x)=e^{-x^{2}}$.

  (i) Calculate the characteristic curves of this equation and show that $u$ remains constant along these curves. Qualitatively sketch the characteristics in the $(x, t)$ diagram, i.e. the $x$ axis is the horizontal and the $t$ axis is the vertical axis.

  (ii) Let $\tilde{x}_{0}$ denote the $x$ value of a characteristic at time $t=0$ and thus label the characteristic curves. Let $\tilde{x}$ denote the $x$ value at time $t$ of a characteristic with given $\tilde{x}_{0}$. Show that $\partial \tilde{x} / \partial \tilde{x}_{0}$ becomes a non-monotonic function of $\tilde{x}_{0}$ (at fixed $t$ ) at times $t>\sqrt{e / 2}$, i.e. $\tilde{x}\left(\tilde{x}_{0}\right)$ has a local minimum or maximum. Qualitatively sketch snapshots of the solution $u(t, x)$ for a few fixed values of $t \in[0, \sqrt{e / 2}]$ and briefly interpret the onset of the non-monotonic behaviour of $\tilde{x}\left(\tilde{x}_{0}\right)$ at $t=\sqrt{e / 2}$.

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• Paper 3, Section I, $3 \mathrm{E}$

  #Metric and Topological Spaces

  2015

  Define what it means for a topological space $X$ to be (i) connected (ii) path-connected.

  Prove that any path-connected space $X$ is connected. [You may assume the interval $[0,1]$ is connected. $]$

  Give a counterexample (without justification) to the converse statement.

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• Paper 2, Section I, $4 \mathbf{E}$

  #Metric and Topological Spaces

  2015

  Let $X$ and $Y$ be topological spaces and $f: X \rightarrow Y$ a continuous map. Suppose $H$ is a subset of $X$ such that $f(\bar{H})$ is closed (where $\bar{H}$ denotes the closure of $H$ ). Prove that $f(\bar{H})=\overline{f(H)} .$

  Give an example where $f, X, Y$ and $H$ are as above but $f(\bar{H})$ is not closed.

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• Paper 1, Section II, E

  #Metric and Topological Spaces

  2015

  Give the definition of a metric on a set $X$ and explain how this defines a topology on $X$.

  Suppose $(X, d)$ is a metric space and $U$ is an open set in $X$. Let $x, y \in X$ and $\epsilon>0$ such that the open ball $B_{\epsilon}(y) \subseteq U$ and $x \in B_{\epsilon / 2}(y)$. Prove that $y \in B_{\epsilon / 2}(x) \subseteq U$.

  Explain what it means (i) for a set $S$ to be dense in $X$, (ii) to say $\mathcal{B}$ is a base for a topology $\mathcal{T}$.

  Prove that any metric space which contains a countable dense set has a countable basis.

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• Paper 4, Section II, E

  #Metric and Topological Spaces

  2015

  Explain what it means for a metric space $(M, d)$ to be (i) compact, (ii) sequentially compact. Prove that a compact metric space is sequentially compact, stating clearly any results that you use.

  Let $(M, d)$ be a compact metric space and suppose $f: M \rightarrow M$ satisfies $d(f(x), f(y))=d(x, y)$ for all $x, y \in M$. Prove that $f$ is surjective, stating clearly any results that you use. [Hint: Consider the sequence $\left(f^{n}(x)\right)$ for $x \in M$.]

  Give an example to show that the result does not hold if $M$ is not compact.

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• Paper 1, Section I, 6D

  #Numerical Analysis

  2015

  Let

  $$A=\left[\begin{array}{rrrr} 1 & 4 & 3 & 2 \\ 4 & 17 & 13 & 11 \\ 3 & 13 & 13 & 12 \\ 2 & 11 & 12 & \lambda \end{array}\right], \quad b=\left[\begin{array}{l} 1 \\ 1 \\ 3 \\ 2 \end{array}\right]$$

  where $\lambda$ is a real parameter. Find the $L U$ factorization of the matrix $A$. Give the constraint on $\lambda$ for A to be positive definite.

  For $\lambda=18$, use this factorization to solve the system $A x=b$ via forward and backward substitution.

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• Paper 4, Section I, D

  #Numerical Analysis

  2015

  Given $n+1$ distinct points $\left\{x_{0}, x_{1}, \ldots, x_{n}\right\}$, let $p_{n} \in \mathbb{P}_{n}$ be the real polynomial of degree $n$ that interpolates a continuous function $f$ at these points. State the Lagrange interpolation formula.

  Prove that $p_{n}$ can be written in the Newton form

  $$p_{n}(x)=f\left(x_{0}\right)+\sum_{k=1}^{n} f\left[x_{0}, \ldots, x_{k}\right] \prod_{i=0}^{k-1}\left(x-x_{i}\right)$$

  where $f\left[x_{0}, \ldots, x_{k}\right]$ is the divided difference, which you should define. [An explicit expression for the divided difference is not required.]

  Explain why it can be more efficient to use the Newton form rather than the Lagrange formula.

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• Paper 1, Section II, 18D

  #Numerical Analysis

  2015

  Determine the real coefficients $b_{1}, b_{2}, b_{3}$ such that

  $$\int_{-2}^{2} f(x) d x=b_{1} f(-1)+b_{2} f(0)+b_{3} f(1)$$

  is exact when $f(x)$ is any real polynomial of degree 2 . Check explicitly that the quadrature is exact for $f(x)=x^{2}$ with these coefficients.

  State the Peano kernel theorem and define the Peano kernel $K(\theta)$. Use this theorem to show that if $f \in C^{3}[-2,2]$, and $b_{1}, b_{2}, b_{3}$ are chosen as above, then

  $$\left|\int_{-2}^{2} f(x) d x-b_{1} f(-1)-b_{2} f(0)-b_{3} f(1)\right| \leqslant \frac{4}{9} \max _{\xi \in[-2,2]}\left|f^{(3)}(\xi)\right|$$

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• Paper 3, Section II, D

  #Numerical Analysis

  2015

  Define the QR factorization of an $m \times n$ matrix $A$. Explain how it can be used to solve the least squares problem of finding the vector $x^{*} \in \mathbb{R}^{n}$ which minimises $\|\mathrm{A} x-b\|$, where $b \in \mathbb{R}^{m}, m>n$, and $\|\cdot\|$ is the Euclidean norm.

  Explain how to construct $Q$ and $R$ by the Gram-Schmidt procedure. Why is this procedure not useful for numerical factorization of large matrices?

  Let

  $$A=\left[\begin{array}{rrr} 5 & 6 & -14 \\ 5 & 4 & 4 \\ -5 & 2 & -8 \\ 5 & 12 & -18 \end{array}\right], \quad b=\left[\begin{array}{l} 1 \\ 1 \\ 1 \\ 0 \end{array}\right]$$

  Using the Gram-Schmidt procedure find a QR decomposition of A. Hence solve the least squares problem giving both $x^{*}$ and $\left\|\mathrm{A} x^{*}-b\right\|$.

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• Paper 2, Section II, D

  #Numerical Analysis

  2015

  Define the linear stability domain for a numerical method to solve $y^{\prime}=f(t, y)$. What is meant by an $A$-stable method? Briefly explain the relevance of these concepts in the numerical solution of ordinary differential equations.

  Consider

  $$y_{n+1}=y_{n}+h\left[\theta f\left(t_{n}, y_{n}\right)+(1-\theta) f\left(t_{n+1}, y_{n+1}\right)\right]$$

  where $\theta \in[0,1]$. What is the order of this method?

  Find the linear stability domain of this method. For what values of $\theta$ is the method A-stable?

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• Paper 1, Section I, H

  #Optimization

  2015

  (a) Consider a network with vertices in $V=\{1, \ldots, n\}$ and directed edges $(i, j)$ in $E \subseteq V \times V$. Suppose that 1 is the source and $n$ is the sink. Let $C_{i j}, 0<C_{i j}<\infty$, be the capacity of the edge from vertex $i$ to vertex $j$ for $(i, j) \in E$. Let a cut be a partition of $V=\{1, \ldots, n\}$ into $S$ and $V \backslash S$ with $1 \in S$ and $n \in V \backslash S$. Define the capacity of the cut $S$. Write down the maximum flow problem. Prove that the maximum flow is bounded above by the minimum cut capacity.

  (b) Find the maximum flow from the source to the sink in the network below, where the directions and capacities of the edges are shown. Explain your reasoning.

  

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• Paper 2, Section I, H

  #Optimization

  2015

  Define what it means to say that a set $S \subseteq \mathbb{R}^{n}$ is convex. What is meant by an extreme point of a convex set $S$ ?

  Consider the set $S \subseteq \mathbb{R}^{2}$ given by

  $$S=\left\{\left(x_{1}, x_{2}\right): x_{1}+4 x_{2} \leqslant 30,3 x_{1}+7 x_{2} \leqslant 60, x_{1} \geqslant 0, x_{2} \geqslant 0\right\}$$

  Show that $S$ is convex, and give the coordinates of all extreme points of $S$.

  For all possible choices of $c_{1}>0$ and $c_{2}>0$, find the maximum value of $c_{1} x_{1}+c_{2} x_{2}$ subject to $\left(x_{1}, x_{2}\right) \in S$.

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• Paper 4, Section II, 20H

  #Optimization

  2015

  Suppose the recycling manager in a particular region is responsible for allocating all the recyclable waste that is collected in $n$ towns in the region to the $m$ recycling centres in the region. Town $i$ produces $s_{i}$ lorry loads of recyclable waste each day, and recycling centre $j$ needs to handle $d_{j}$ lorry loads of waste a day in order to be viable. Suppose that $\sum_{i} s_{i}=\sum_{j} d_{j}$. Suppose further that $c_{i j}$ is the cost of transporting a lorry load of waste from town $i$ to recycling centre $j$. The manager wishes to decide the number $x_{i j}$ of lorry loads of recyclable waste that should go from town $i$ to recycling centre $j$, $i=1, \ldots, n, j=1, \ldots, m$, in such a way that all the recyclable waste produced by each town is transported to recycling centres each day, and each recycling centre works exactly at the viable level each day. Use the Lagrangian sufficiency theorem, which you should quote carefully, to derive necessary and sufficient conditions for $\left(x_{i j}\right)$ to minimise the total cost under the above constraints.

  Suppose that there are three recycling centres $A, B$ and $C$, needing 5,20 and 20 lorry loads of waste each day, respectively, and suppose there are three towns $a, b$ and $c$ producing 20,15 and 10 lorry loads of waste each day, respectively. The costs of transporting a lorry load of waste from town $a$ to recycling centres $A, B$ and $C$ are $£ 90, £ 100$ and $£ 100$, respectively. The corresponding costs for town $b$ are $£ 130, £ 140$ and $£ 100$, while for town $c$ they are $£ 110, £ 80$ and $£ 80$. Recycling centre $A$ has reported that it currently receives 5 lorry loads of waste per day from town $a$, and recycling centre $C$ has reported that it currently receives 10 lorry loads of waste per day from each of towns $b$ and c. Recycling centre $B$ has failed to report. What is the cost of the current arrangement for transporting waste from the towns to the recycling centres? Starting with the current arrangement as an initial solution, use the transportation algorithm (explaining each step carefully) in order to advise the recycling manager how many lorry loads of waste should go from each town to each of the recycling centres in order to minimise the cost. What is the minimum cost?

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• Paper 3, Section II, H

  #Optimization

  2015

  Consider the linear programming problem $P$ :

  $$\text { minimise } c^{T} x \text { subject to } A x \geqslant b, x \geqslant 0,$$

  where $x$ and $c$ are in $\mathbb{R}^{n}, A$ is a real $m \times n$ matrix, $b$ is in $\mathbb{R}^{m}$ and ${ }^{T}$ denotes transpose. Derive the dual linear programming problem $D$. Show from first principles that the dual of $D$ is $P$.

  Suppose that $c^{T}=(6,10,11), b^{T}=(1,1,3)$ and $A=\left(\begin{array}{lll}1 & 3 & 8 \\ 1 & 1 & 2 \\ 2 & 4 & 4\end{array}\right)$. Write down the dual $D$ and find the optimal solution of the dual using the simplex algorithm. Hence, or otherwise, find the optimal solution $x^{*}=\left(x_{1}^{*}, x_{2}^{*}, x_{3}^{*}\right)$ of $P$.

  Suppose that $c$ is changed to $\tilde{c}=\left(6+\varepsilon_{1}, 10+\varepsilon_{2}, 11+\varepsilon_{3}\right)$. Give necessary and sufficient conditions for $x^{*}$ still to be the optimal solution of $P$. If $\varepsilon_{1}=\varepsilon_{2}=0$, find the range of values for $\varepsilon_{3}$ for which $x^{*}$ is still the optimal solution of $P$.

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• Paper 4, Section I, D

  #Quantum Mechanics

  2015

  The radial wavefunction $R(r)$ for an electron in a hydrogen atom satisfies the equation

  $$-\frac{\hbar^{2}}{2 m r^{2}} \frac{d}{d r}\left(r^{2} \frac{d}{d r} R(r)\right)+\frac{\hbar^{2}}{2 m r^{2}} \ell(\ell+1) R(r)-\frac{e^{2}}{4 \pi \epsilon_{0} r} R(r)=E R(r)$$

  Briefly explain the origin of each term in this equation.

  The wavefunctions for the ground state and the first radially excited state, both with $\ell=0$, can be written as

  $$\begin{aligned} &R_{1}(r)=N_{1} e^{-\alpha r} \\ &R_{2}(r)=N_{2}\left(1-\frac{1}{2} r \alpha\right) e^{-\frac{1}{2} \alpha r} \end{aligned}$$

  where $N_{1}$ and $N_{2}$ are normalisation constants. Verify that $R_{1}(r)$ is a solution of $(*)$, determining $\alpha$ and finding the corresponding energy eigenvalue $E_{1}$. Assuming that $R_{2}(r)$ is a solution of $(*)$, compare coefficients of the dominant terms when $r$ is large to determine the corresponding energy eigenvalue $E_{2}$. [You do not need to find $N_{1}$ or $N_{2}$, nor show that $R_{2}$ is a solution of $\left.(*) .\right]$

  A hydrogen atom makes a transition from the first radially excited state to the ground state, emitting a photon. What is the angular frequency of the emitted photon?

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• Paper 3, Section $I$, D

  #Quantum Mechanics

  2015

  A quantum-mechanical system has normalised energy eigenstates $\chi_{1}$ and $\chi_{2}$ with non-degenerate energies $E_{1}$ and $E_{2}$ respectively. The observable $A$ has normalised eigenstates,

  $$\begin{aligned} \phi_{1} &=C\left(\chi_{1}+2 \chi_{2}\right), & & \text { eigenvalue }=a_{1} \\ \phi_{2} &=C\left(2 \chi_{1}-\chi_{2}\right), & & \text { eigenvalue }=a_{2} \end{aligned}$$

  where $C$ is a positive real constant. Determine $C$.

  Initially, at time $t=0$, the state of the system is $\phi_{1}$. Write down an expression for $\psi(t)$, the state of the system with $t \geqslant 0$. What is the probability that a measurement of energy at time $t$ will yield $E_{2}$ ?

  For the same initial state, determine the probability that a measurement of $A$ at time $t>0$ will yield $a_{1}$ and the probability that it will yield $a_{2}$.

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• Paper 1, Section II, D

  #Quantum Mechanics

  2015

  Write down expressions for the probability density $\rho(x, t)$ and the probability current $j(x, t)$ for a particle in one dimension with wavefunction $\Psi(x, t)$. If $\Psi(x, t)$ obeys the timedependent Schrödinger equation with a real potential, show that

  $$\frac{\partial j}{\partial x}+\frac{\partial \rho}{\partial t}=0$$

  Consider a stationary state, $\Psi(x, t)=\psi(x) e^{-i E t / \hbar}$, with

  $$\psi(x) \sim \begin{cases}e^{i k_{1} x}+R e^{-i k_{1} x} & x \rightarrow-\infty \\ T e^{i k_{2} x} & x \rightarrow+\infty\end{cases}$$

  where $E, k_{1}, k_{2}$ are real. Evaluate $j(x, t)$ for this state in the regimes $x \rightarrow+\infty$ and $x \rightarrow-\infty$.

  Consider a real potential,

  $$V(x)=-\alpha \delta(x)+U(x), \quad U(x)= \begin{cases}0 & x<0 \\ V_{0} & x>0\end{cases}$$

  where $\delta(x)$ is the Dirac delta function, $V_{0}>0$ and $\alpha>0$. Assuming that $\psi(x)$ is continuous at $x=0$, derive an expression for

  $$\lim _{\epsilon \rightarrow 0}\left[\psi^{\prime}(\epsilon)-\psi^{\prime}(-\epsilon)\right]$$

  Hence calculate the reflection and transmission probabilities for a particle incident from $x=-\infty$ with energy $E>V_{0} .$

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• Paper 3, Section II, D

  #Quantum Mechanics

  2015

  Define the angular momentum operators $\hat{L}_{i}$ for a particle in three dimensions in terms of the position and momentum operators $\hat{x}_{i}$ and $\hat{p}_{i}=-i \hbar \frac{\partial}{\partial x_{i}}$. Write down an expression for $\left[\hat{L}_{i}, \hat{L}_{j}\right]$ and use this to show that $\left[\hat{L}^{2}, \hat{L}_{i}\right]=0$ where $\hat{L}^{2}=\hat{L}_{x}^{2}+\hat{L}_{y}^{2}+\hat{L}_{z}^{2}$. What is the significance of these two commutation relations?

  Let $\psi(x, y, z)$ be both an eigenstate of $\hat{L}_{z}$ with eigenvalue zero and an eigenstate of $\hat{L}^{2}$ with eigenvalue $\hbar^{2} l(l+1)$. Show that $\left(\hat{L}_{x}+i \hat{L}_{y}\right) \psi$ is also an eigenstate of both $\hat{L}_{z}$ and $\hat{L}^{2}$ and determine the corresponding eigenvalues.

  Find real constants $A$ and $B$ such that

  $$\phi(x, y, z)=\left(A z^{2}+B y^{2}-r^{2}\right) e^{-r}, \quad r^{2}=x^{2}+y^{2}+z^{2},$$

  is an eigenfunction of $\hat{L}_{z}$ with eigenvalue zero and an eigenfunction of $\hat{L}^{2}$ with an eigenvalue which you should determine. [Hint: You might like to show that $\left.\hat{L}_{i} f(r)=0 .\right]$

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• Paper 2, Section II, D

  #Quantum Mechanics

  2015

  A quantum-mechanical harmonic oscillator has Hamiltonian

  $$\hat{H}=\frac{\hat{p}^{2}}{2}+\frac{1}{2} k^{2} \hat{x}^{2}$$

  where $k$ is a positive real constant. Show that $\hat{x}=x$ and $\hat{p}=-i \hbar \frac{\partial}{\partial x}$ are Hermitian operators.

  The eigenfunctions of $(*)$ can be written as

  $$\psi_{n}(x)=h_{n}(x \sqrt{k / \hbar}) \exp \left(-\frac{k x^{2}}{2 \hbar}\right),$$

  where $h_{n}$ is a polynomial of degree $n$ with even (odd) parity for even (odd) $n$ and $n=0,1,2, \ldots$. Show that $\langle\hat{x}\rangle=\langle\hat{p}\rangle=0$ for all of the states $\psi_{n}$.

  State the Heisenberg uncertainty principle and verify it for the state $\psi_{0}$ by computing $(\Delta x)$ and $(\Delta p)$. [Hint: You should properly normalise the state.]

  The oscillator is in its ground state $\psi_{0}$ when the potential is suddenly changed so that $k \rightarrow 4 k$. If the wavefunction is expanded in terms of the energy eigenfunctions of the new Hamiltonian, $\phi_{n}$, what can be said about the coefficient of $\phi_{n}$ for odd $n$ ? What is the probability that the particle is in the new ground state just after the change?

  [Hint: You may assume that if $I_{n}=\int_{-\infty}^{\infty} e^{-a x^{2}} x^{n} d x$ then $I_{0}=\sqrt{\frac{\pi}{a}}$ and $I_{2}=\frac{1}{2 a} \sqrt{\frac{\pi}{a}}$.]

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• Paper 1, Section I, H

  #Statistics

  2015

  Suppose that $X_{1}, \ldots, X_{n}$ are independent normally distributed random variables, each with mean $\mu$ and variance 1 , and consider testing $H_{0}: \mu=0$ against $H_{1}: \mu=1$. Explain what is meant by the critical region, the size and the power of a test.

  For $0<\alpha<1$, derive the test that is most powerful among all tests of size at most $\alpha$. Obtain an expression for the power of your test in terms of the standard normal distribution function $\Phi(\cdot)$.

  [Results from the course may be used without proof provided they are clearly stated.]

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• Paper 2, Section I, H

  #Statistics

  2015

  Suppose that, given $\theta$, the random variable $X$ has $\mathbb{P}(X=k)=e^{-\theta} \theta^{k} / k !$, $k=0,1,2, \ldots .$ Suppose that the prior density of $\theta$ is $\pi(\theta)=\lambda e^{-\lambda \theta}, \theta>0$, for some known $\lambda(>0)$. Derive the posterior density $\pi(\theta \mid x)$ of $\theta$ based on the observation $X=x$.

  For a given loss function $L(\theta, a)$, a statistician wants to calculate the value of $a$ that minimises the expected posterior loss

  $$\int L(\theta, a) \pi(\theta \mid x) d \theta$$

  Suppose that $x=0$. Find $a$ in terms of $\lambda$ in the following cases:

  (a) $L(\theta, a)=(\theta-a)^{2}$;

  (b) $L(\theta, a)=|\theta-a|$.

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• Paper 4, Section II, H

  #Statistics

  2015

  Consider a linear model $\mathbf{Y}=X \boldsymbol{\beta}+\varepsilon$ where $\mathbf{Y}$ is an $n \times 1$ vector of observations, $X$ is a known $n \times p$ matrix, $\boldsymbol{\beta}$ is a $p \times 1(p<n)$ vector of unknown parameters and $\varepsilon$ is an $n \times 1$ vector of independent normally distributed random variables each with mean zero and unknown variance $\sigma^{2}$. Write down the log-likelihood and show that the maximum likelihood estimators $\hat{\boldsymbol{\beta}}$ and $\hat{\sigma}^{2}$ of $\boldsymbol{\beta}$ and $\sigma^{2}$ respectively satisfy

  $$X^{T} X \hat{\boldsymbol{\beta}}=X^{T} \mathbf{Y}, \quad \frac{1}{\hat{\sigma}^{4}}(\mathbf{Y}-X \hat{\boldsymbol{\beta}})^{T}(\mathbf{Y}-X \hat{\boldsymbol{\beta}})=\frac{n}{\hat{\sigma}^{2}}$$

  $(T$ denotes the transpose $)$. Assuming that $X^{T} X$ is invertible, find the solutions $\hat{\boldsymbol{\beta}}$ and $\hat{\sigma}^{2}$ of these equations and write down their distributions.

  Prove that $\hat{\boldsymbol{\beta}}$ and $\hat{\sigma}^{2}$ are independent.

  Consider the model $Y_{i j}=\mu_{i}+\gamma x_{i j}+\varepsilon_{i j}, i=1,2,3$ and $j=1,2,3$. Suppose that, for all $i, x_{i 1}=-1, x_{i 2}=0$ and $x_{i 3}=1$, and that $\varepsilon_{i j}, i, j=1,2,3$, are independent $N\left(0, \sigma^{2}\right)$ random variables where $\sigma^{2}$ is unknown. Show how this model may be written as a linear model and write down $\mathbf{Y}, X, \boldsymbol{\beta}$ and $\varepsilon$. Find the maximum likelihood estimators of $\mu_{i}$ $(i=1,2,3), \gamma$ and $\sigma^{2}$ in terms of the $Y_{i j}$. Derive a $100(1-\alpha) \%$ confidence interval for $\sigma^{2}$ and for $\mu_{2}-\mu_{1}$.

  [You may assume that, if $\mathbf{W}=\left(\mathbf{W}_{1}^{T}, \mathbf{W}_{2}^{T}\right)^{T}$ is multivariate normal with $\operatorname{cov}\left(\mathbf{W}_{1}, \mathbf{W}_{2}\right)=0$, then $\mathbf{W}_{1}$ and $\mathbf{W}_{2}$ are independent.]

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• Paper 1, Section II, H

  #Statistics

  2015

  Suppose $X_{1}, \ldots, X_{n}$ are independent identically distributed random variables each with probability mass function $\mathbb{P}\left(X_{i}=x_{i}\right)=p\left(x_{i} ; \theta\right)$, where $\theta$ is an unknown parameter. State what is meant by a sufficient statistic for $\theta$. State the factorisation criterion for a sufficient statistic. State and prove the Rao-Blackwell theorem.

  Suppose that $X_{1}, \ldots, X_{n}$ are independent identically distributed random variables with

  $$\mathbb{P}\left(X_{i}=x_{i}\right)=\left(\begin{array}{c} m \\ x_{i} \end{array}\right) \theta^{x_{i}}(1-\theta)^{m-x_{i}}, \quad x_{i}=0, \ldots, m$$

  where $m$ is a known positive integer and $\theta$ is unknown. Show that $\tilde{\theta}=X_{1} / m$ is unbiased for $\theta$.

  Show that $T=\sum_{i=1}^{n} X_{i}$ is sufficient for $\theta$ and use the Rao-Blackwell theorem to find another unbiased estimator $\hat{\theta}$ for $\theta$, giving details of your derivation. Calculate the variance of $\hat{\theta}$ and compare it to the variance of $\tilde{\theta}$.

  A statistician cannot remember the exact statement of the Rao-Blackwell theorem and calculates $\mathbb{E}\left(T \mid X_{1}\right)$ in an attempt to find an estimator of $\theta$. Comment on the suitability or otherwise of this approach, giving your reasons.

  [Hint: If $a$ and $b$ are positive integers then, for $r=0,1, \ldots, a+b,\left(\begin{array}{c}a+b \\ r\end{array}\right)=$ $\left.\sum_{j=0}^{r}\left(\begin{array}{c}a \\ j\end{array}\right)\left(\begin{array}{c}b \\ r-j\end{array}\right) .\right]$

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• Paper 3, Section II, H

  #Statistics

  2015

  (a) Suppose that $X_{1}, \ldots, X_{n}$ are independent identically distributed random variables, each with density $f(x)=\theta \exp (-\theta x), x>0$ for some unknown $\theta>0$. Use the generalised likelihood ratio to obtain a size $\alpha$ test of $H_{0}: \theta=1$ against $H_{1}: \theta \neq 1$.

  (b) A die is loaded so that, if $p_{i}$ is the probability of face $i$, then $p_{1}=p_{2}=\theta_{1}$, $p_{3}=p_{4}=\theta_{2}$ and $p_{5}=p_{6}=\theta_{3}$. The die is thrown $n$ times and face $i$ is observed $x_{i}$ times. Write down the likelihood function for $\theta=\left(\theta_{1}, \theta_{2}, \theta_{3}\right)$ and find the maximum likelihood estimate of $\theta$.

  Consider testing whether or not $\theta_{1}=\theta_{2}=\theta_{3}$ for this die. Find the generalised likelihood ratio statistic $\Lambda$ and show that

  $$2 \log _{e} \Lambda \approx T, \quad \text { where } T=\sum_{i=1}^{3} \frac{\left(o_{i}-e_{i}\right)^{2}}{e_{i}}$$

  where you should specify $o_{i}$ and $e_{i}$ in terms of $x_{1}, \ldots, x_{6}$. Explain how to obtain an approximate size $0.05$ test using the value of $T$. Explain what you would conclude (and why ) if $T=2.03$.

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• Paper 1, Section I, A

  #Variational Principles

  2015

  Consider a frictionless bead on a stationary wire. The bead moves under the action of gravity acting in the negative $y$-direction and the wire traces out a path $y(x)$, connecting points $(x, y)=(0,0)$ and $(1,0)$. Using a first integral of the Euler-Lagrange equations, find the choice of $y(x)$ which gives the shortest travel time, starting from rest. You may give your solution for $y$ and $x$ separately, in parametric form.

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• Paper 3, Section $I$, A

  #Variational Principles

  2015

  (a) Define what it means for a function $f: \mathbb{R}^{n} \rightarrow \mathbb{R}$ to be convex.

  (b) Define the Legendre transform $f^{*}(p)$ of a convex function $f(x)$, where $x \in \mathbb{R}$. Show that $f^{*}(p)$ is a convex function.

  (c) Find the Legendre transform $f^{*}(p)$ of the function $f(x)=e^{x}$, and the domain of $f^{*}$.

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• Paper 4, Section II, F

  #Algebraic Geometry

  2015

  (i) Explain how a linear system on a curve $C$ may induce a morphism from $C$ to projective space. What condition on the linear system is necessary to yield a morphism $f: C \rightarrow \mathbb{P}^{n}$ such that the pull-back of a hyperplane section is an element of the linear system? What condition is necessary to imply the morphism is an embedding?

  (ii) State the Riemann-Roch theorem for curves.

  (iii) Show that any divisor of degree 5 on a curve $C$ of genus 2 induces an embedding.

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• Paper 3, Section II, F

  #Algebraic Geometry

  2015

  (i) Let $X$ be an affine variety. Define the tangent space of $X$ at a point $P$. Say what it means for the variety to be singular at $P$.

  (ii) Find the singularities of the surface in $\mathbb{P}^{3}$ given by the equation

  $$x y z+y z w+z w x+w x y=0 .$$

  (iii) Consider $C=Z\left(x^{2}-y^{3}\right) \subseteq \mathbb{A}^{2}$. Let $X \rightarrow \mathbb{A}^{2}$ be the blowup of the origin. Compute the proper transform of $C$ in $X$, and show it is non-singular.

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• Paper 2, Section II, F

  #Algebraic Geometry

  2015

  (i) Define the radical of an ideal.

  (ii) Assume the following statement: If $k$ is an algebraically closed field and $I \subseteq$ $k\left[x_{1}, \ldots, x_{n}\right]$ is an ideal, then either $I=(1)$ or $Z(I) \neq \emptyset$. Prove the Hilbert Nullstellensatz, namely that if $I \subseteq k\left[x_{1}, \ldots, x_{n}\right]$ with $k$ algebraically closed, then

  $$I(Z(I))=\sqrt{I}$$

  (iii) Show that if $A$ is a commutative ring and $I, J \subseteq A$ are ideals, then

  $$\sqrt{I \cap J}=\sqrt{I} \cap \sqrt{J} .$$

  (iv) Is

  $$\sqrt{I+J}=\sqrt{I}+\sqrt{J} ?$$

  Give a proof or a counterexample.

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• Paper 1, Section II, F

  #Algebraic Geometry

  2015

  Let $k$ be an algebraically closed field.

  (i) Let $X$ and $Y$ be affine varieties defined over $k$. Given a map $f: X \rightarrow Y$, define what it means for $f$ to be a morphism of affine varieties.

  (ii) With $X, Y$ still affine varieties over $k$, show that there is a one-to-one correspondence between $\operatorname{Hom}(X, Y)$, the set of morphisms between $X$ and $Y$, and $\operatorname{Hom}(A(Y), A(X))$, the set of $k$-algebra homomorphisms between $A(Y)$ and $A(X)$.

  (iii) Let $f: \mathbb{A}^{2} \rightarrow \mathbb{A}^{4}$ be given by $f(t, u)=\left(u, t, t^{2}, t u\right)$. Show that the image of $f$ is an affine variety $X$, and find a set of generators for $I(X)$.

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• Paper 3, Section II, H

  #Algebraic Topology

  2015

  Let $K$ and $L$ be simplicial complexes. Explain what is meant by a simplicial approximation to a continuous map $f:|K| \rightarrow|L|$. State the simplicial approximation theorem, and define the homomorphism induced on homology by a continuous map between triangulable spaces. [You do not need to show that the homomorphism is welldefined.]

  Let $h: S^{1} \rightarrow S^{1}$ be given by $z \mapsto z^{n}$ for a positive integer $n$, where $S^{1}$ is considered as the unit complex numbers. Compute the map induced by $h$ on homology.

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• Paper 4, Section II, H

  #Algebraic Topology

  2015

  State the Mayer-Vietoris theorem for a simplicial complex $K$ which is the union of two subcomplexes $M$ and $N$. Explain briefly how the connecting homomorphism $\partial_{n}: H_{n}(K) \rightarrow H_{n-1}(M \cap N)$ is defined.

  If $K$ is the union of subcomplexes $M_{1}, M_{2}, \ldots, M_{n}$, with $n \geqslant 2$, such that each intersection

  $$M_{i_{1}} \cap M_{i_{2}} \cap \cdots \cap M_{i_{k}}, \quad 1 \leqslant k \leqslant n,$$

  is either empty or has the homology of a point, then show that

  $$H_{i}(K)=0 \quad \text { for } \quad i \geqslant n-1 .$$

  Construct examples for each $n \geqslant 2$ showing that this is sharp.

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• Paper 2, Section II, H

  #Algebraic Topology

  2015

  Define what it means for $p: \widetilde{X} \rightarrow X$ to be a covering map, and what it means to say that $p$ is a universal cover.

  Let $p: \tilde{X} \rightarrow X$ be a universal cover, $A \subset X$ be a locally path connected subspace, and $\tilde{A} \subset p^{-1}(A)$ be a path component containing a point $\tilde{a}_{0}$ with $p\left(\tilde{a}_{0}\right)=a_{0}$. Show that the restriction $\left.p\right|_{\tilde{A}}: \widetilde{A} \rightarrow A$ is a covering map, and that under the Galois correspondence it corresponds to the subgroup

  $$\operatorname{Ker}\left(\pi_{1}\left(A, a_{0}\right) \rightarrow \pi_{1}\left(X, a_{0}\right)\right)$$

  of $\pi_{1}\left(A, a_{0}\right)$.

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• Paper 1, Section II, H

  #Algebraic Topology

  2015

  State carefully a version of the Seifert-van Kampen theorem for a cover of a space by two closed sets.

  Let $X$ be the space obtained by gluing together a Möbius band $M$ and a torus $T=S^{1} \times S^{1}$ along a homeomorphism of the boundary of $M$ with $S^{1} \times\{1\} \subset T$. Find a presentation for the fundamental group of $X$, and hence show that it is infinite and non-abelian.

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• Paper 4, Section II,

  #Applications of Quantum Mechanics

  2015

  Let $\Lambda$ be a Bravais lattice with basis vectors $\mathbf{a}_{1}, \mathbf{a}_{2}, \mathbf{a}_{3}$. Define the reciprocal lattice $\Lambda^{*}$ and write down basis vectors $\mathbf{b}_{1}, \mathbf{b}_{2}, \mathbf{b}_{3}$ for $\Lambda^{*}$ in terms of the basis for $\Lambda$.

  A finite crystal consists of identical atoms at sites of $\Lambda$ given by

  $$\ell=n_{1} \mathbf{a}_{1}+n_{2} \mathbf{a}_{2}+n_{3} \mathbf{a}_{3} \quad \text { with } \quad 0 \leqslant n_{i}<N_{i}$$

  A particle of mass $m$ scatters off the crystal; its wavevector is $\mathbf{k}$ before scattering and $\mathbf{k}^{\prime}$ after scattering, with $|\mathbf{k}|=\left|\mathbf{k}^{\prime}\right|$. Show that the scattering amplitude in the Born approximation has the form

  $$-\frac{m}{2 \pi \hbar^{2}} \Delta(\mathbf{q}) \tilde{U}(\mathbf{q}), \quad \mathbf{q}=\mathbf{k}^{\prime}-\mathbf{k}$$

  where $U(\mathbf{x})$ is the potential due to a single atom at the origin and $\Delta(\mathbf{q})$ depends on the crystal structure. [You may assume that in the Born approximation the amplitude for scattering off a potential $V(\mathbf{x})$ is $-\left(m / 2 \pi \hbar^{2}\right) \tilde{V}(\mathbf{q})$ where tilde denotes the Fourier transform.]

  Derive an expression for $|\Delta(\mathbf{q})|$ that is valid when $e^{-i \mathbf{q} \cdot \mathbf{a}_{i}} \neq 1$. Show also that when $\mathbf{q}$ is a reciprocal lattice vector $|\Delta(\mathbf{q})|$ is equal to the total number of atoms in the crystal. Comment briefly on the significance of these results.

  Now suppose that $\Lambda$ is a face-centred-cubic lattice:

  $$\mathbf{a}_{1}=\frac{a}{2}(\hat{\mathbf{y}}+\hat{\mathbf{z}}), \quad \mathbf{a}_{2}=\frac{a}{2}(\hat{\mathbf{z}}+\hat{\mathbf{x}}), \quad \mathbf{a}_{3}=\frac{a}{2}(\hat{\mathbf{x}}+\hat{\mathbf{y}})$$

  where $a$ is a constant. Show that for a particle incident with $|\mathbf{k}|>2 \pi / a$, enhanced scattering is possible for at least two values of the scattering angle, $\theta_{1}$ and $\theta_{2}$, related by

  $$\frac{\sin \left(\theta_{1} / 2\right)}{\sin \left(\theta_{2} / 2\right)}=\frac{\sqrt{3}}{2}$$

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• Paper 1, Section II, A

  #Applications of Quantum Mechanics

  2015

  Define the Rayleigh-Ritz quotient $R[\psi]$ for a normalisable state $|\psi\rangle$ of a quantum system with Hamiltonian $H$. Given that the spectrum of $H$ is discrete and that there is a unique ground state of energy $E_{0}$, show that $R[\psi] \geqslant E_{0}$ and that equality holds if and only if $|\psi\rangle$ is the ground state.

  A simple harmonic oscillator (SHO) is a particle of mass $m$ moving in one dimension subject to the potential

  $$V(x)=\frac{1}{2} m \omega^{2} x^{2}$$

  Estimate the ground state energy $E_{0}$ of the SHO by using the ground state wavefunction for a particle in an infinite potential well of width $a$, centred on the origin (the potential is $U(x)=0$ for $|x|<a / 2$ and $U(x)=\infty$ for $|x|>a / 2)$. Take $a$ as the variational parameter.

  Perform a similar estimate for the energy $E_{1}$ of the first excited state of the SHO by using the first excited state of the infinite potential well as a trial wavefunction.

  Is the estimate for $E_{1}$ necessarily an upper bound? Justify your answer.

  $\left[\right.$ You may use : $\int_{-\pi / 2}^{\pi / 2} y^{2} \cos ^{2} y d y=\frac{\pi}{4}\left(\frac{\pi^{2}}{6}-1\right) \quad$ and $\left.\quad \int_{-\pi}^{\pi} y^{2} \sin ^{2} y d y=\pi\left(\frac{\pi^{2}}{3}-\frac{1}{2}\right) \cdot\right]$

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• Paper 3, Section II, A

  #Applications of Quantum Mechanics

  2015

  A particle of mass $m$ and energy $E=-\hbar^{2} \kappa^{2} / 2 m<0$ moves in one dimension subject to a periodic potential

  $$V(x)=-\frac{\hbar^{2} \lambda}{m} \sum_{\ell=-\infty}^{\infty} \delta(x-\ell a) \quad \text { with } \quad \lambda>0 .$$

  Determine the corresponding Floquet matrix $\mathcal{M}$. [You may assume without proof that for the Schrödinger equation with potential $\alpha \delta(x)$ the wavefunction $\psi(x)$ is continuous at $x=0$ and satisfies $\left.\psi^{\prime}(0+)-\psi^{\prime}(0-)=\left(2 m \alpha / \hbar^{2}\right) \psi(0) .\right]$

  Explain briefly, with reference to Bloch's theorem, how restrictions on the energy of a Bloch state can be derived from $\mathcal{M}$. Deduce that for the potential $V(x)$ above, $\kappa$ is confined to a range whose boundary values are determined by

  $$\tanh \left(\frac{\kappa a}{2}\right)=\frac{\kappa}{\lambda} \quad \text { and } \quad \operatorname{coth}\left(\frac{\kappa a}{2}\right)=\frac{\kappa}{\lambda} .$$

  Sketch the left-hand and right-hand sides of each of these equations as functions of $y=\kappa a / 2$. Hence show that there is exactly one allowed band of negative energies with either (i) $E_{-} \leqslant E<0$ or (ii) $E_{-} \leqslant E \leqslant E_{+}<0$ and determine the values of $\lambda a$ for which each of these cases arise. [You should not attempt to evaluate the constants $E_{\pm} .$]

  Comment briefly on the limit $a \rightarrow \infty$ with $\lambda$ fixed.

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• Paper 2, Section II, A

  #Applications of Quantum Mechanics

  2015

  A beam of particles of mass $m$ and energy $\hbar^{2} k^{2} / 2 m$ is incident on a target at the origin described by a spherically symmetric potential $V(r)$. Assuming the potential decays rapidly as $r \rightarrow \infty$, write down the asymptotic form of the wavefunction, defining the scattering amplitude $f(\theta)$.

  Consider a free particle with energy $\hbar^{2} k^{2} / 2 m$. State, without proof, the general axisymmetric solution of the Schrödinger equation for $r>0$ in terms of spherical Bessel and Neumann functions $j_{\ell}$ and $n_{\ell}$, and Legendre polynomials $P_{\ell}(\ell=0,1,2, \ldots)$. Hence define the partial wave phase shifts $\delta_{\ell}$ for scattering from a potential $V(r)$ and derive the partial wave expansion for $f(\theta)$ in terms of phase shifts.

  Now suppose

  $$V(r)=\left\{\begin{array}{cc} \hbar^{2} \gamma^{2} / 2 m & r<a \\ 0 & r>a \end{array}\right.$$

  with $\gamma>k$. Show that the S-wave phase shift $\delta_{0}$ obeys

  $$\frac{\tanh (\kappa a)}{\kappa a}=\frac{\tan \left(k a+\delta_{0}\right)}{k a}$$

  where $\kappa^{2}=\gamma^{2}-k^{2}$. Deduce that for an S-wave solution

  $$f \rightarrow \frac{\tanh \gamma a-\gamma a}{\gamma} \quad \text { as } \quad k \rightarrow 0$$

  [You may assume : $\quad \exp (i k r \cos \theta)=\sum_{\ell=0}^{\infty}(2 \ell+1) i^{\ell} j_{\ell}(k r) P_{\ell}(\cos \theta)$

  and $j_{\ell}(\rho) \sim \frac{1}{\rho} \sin (\rho-\ell \pi / 2), \quad n_{\ell}(\rho) \sim-\frac{1}{\rho} \cos (\rho-\ell \pi / 2) \quad$ as $\left.\quad \rho \rightarrow \infty .\right]$

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• Paper 4, Section II, K

  #Applied Probability

  2015

  (i) Let $X$ be a Markov chain on $S$ and $A \subset S$. Let $T_{A}$ be the hitting time of $A$ and $\tau_{y}$ denote the total time spent at $y \in S$ by the chain before hitting $A$. Show that if $h(x)=\mathbb{P}_{x}\left(T_{A}<\infty\right)$, then $\mathbb{E}_{x}\left[\tau_{y} \mid T_{A}<\infty\right]=[h(y) / h(x)] \mathbb{E}_{x}\left(\tau_{y}\right) .$

  (ii) Define the Moran model and show that if $X_{t}$ is the number of individuals carrying allele $a$ at time $t \geqslant 0$ and $\tau$ is the fixation time of allele $a$, then

  $$\mathbb{P}\left(X_{\tau}=N \mid X_{0}=i\right)=\frac{i}{N}$$

  Show that conditionally on fixation of an allele $a$ being present initially in $i$ individuals,

  $$\mathbb{E}[\tau \mid \text { fixation }]=N-i+\frac{N-i}{i} \sum_{j=1}^{i-1} \frac{j}{N-j}$$

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• Paper 3, Section II, K

  #Applied Probability

  2015

  (i) Let $X$ be a Poisson process of parameter $\lambda$. Let $Y$ be obtained by taking each point of $X$ and, independently of the other points, keeping it with probability $p$. Show that $Y$ is another Poisson process and find its intensity. Show that for every fixed $t$ the random variables $Y_{t}$ and $X_{t}-Y_{t}$ are independent.

  (ii) Suppose we have $n$ bins, and balls arrive according to a Poisson process of rate 1 . Upon arrival we choose a bin uniformly at random and place the ball in it. We let $M_{n}$ be the maximum number of balls in any bin at time $n$. Show that

  $$\mathbb{P}\left(M_{n} \geqslant(1+\epsilon) \frac{\log n}{\log \log n}\right) \rightarrow 0 \quad \text { as } n \rightarrow \infty$$

  [You may use the fact that if $\xi$ is a Poisson random variable of mean 1 , then

  $$\mathbb{P}(\xi \geqslant x) \leqslant \exp (x-x \log x) .]$$

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• Paper 2, Section II, $24 K$

  #Applied Probability

  2015

  (i) Defne a Poisson process on $\mathbb{R}_{+}$with rate $\lambda$. Let $N$ and $M$ be two independent Poisson processes on $\mathbb{R}_{+}$of rates $\lambda$ and $\mu$ respectively. Prove that $N+M$ is also a Poisson process and find its rate.

  (ii) Let $X$ be a discrete time Markov chain with transition matrix $K$ on the finite state space $S$. Find the generator of the continuous time Markov chain $Y_{t}=X_{N_{t}}$ in terms of $K$ and $\lambda$. Show that if $\pi$ is an invariant distribution for $X$, then it is also invariant for $Y$.

  Suppose that $X$ has an absorbing state $a$. If $\tau_{a}$ and $T_{a}$ are the absorption times for $X$ and $Y$ respectively, write an equation that relates $\mathbb{E}_{x}\left[\tau_{a}\right]$ and $\mathbb{E}_{x}\left[T_{a}\right]$, where $x \in S$.

  [Hint: You may want to prove that if $\xi_{1}, \xi_{2}, \ldots$ are i.i.d. non-negative random variables with $\mathbb{E}\left[\xi_{1}\right]<\infty$ and $M$ is an independent non-negative random variable, then $\left.\mathbb{E}\left[\sum_{i=1}^{M} \xi_{i}\right]=\mathbb{E}[M] \mathbb{E}\left[\xi_{1}\right] .\right]$

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• Paper 1, Section II, K

  #Applied Probability

  2015

  (a) Give the definition of a birth and death chain in terms of its generator. Show that a measure $\pi$ is invariant for a birth and death chain if and only if it solves the detailed balance equations.

  (b) There are $s$ servers in a post office and a single queue. Customers arrive as a Poisson process of rate $\lambda$ and the service times at each server are independent and exponentially distributed with parameter $\mu$. Let $X_{t}$ denote the number of customers in the post office at time $t$. Find conditions on $\lambda, \mu$ and $s$ for $X$ to be positive recurrent, null recurrent and transient, justifying your answers.

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• Paper 4, Section II, C

  #Asymptotic Methods

  2015

  Consider the ordinary differential equation

  $$\frac{d^{2} u}{d z^{2}}+f(z) \frac{d u}{d z}+g(z) u=0$$

  where

  $$f(z) \sim \sum_{m=0}^{\infty} \frac{f_{m}}{z^{m}}, \quad g(z) \sim \sum_{m=0}^{\infty} \frac{g_{m}}{z^{m}}, \quad z \rightarrow \infty$$

  and $f_{m}, g_{m}$ are constants. Look for solutions in the asymptotic form

  $$u(z)=e^{\lambda z} z^{\mu}\left[1+\frac{a}{z}+\frac{b}{z^{2}}+O\left(\frac{1}{z^{3}}\right)\right], \quad z \rightarrow \infty$$

  and determine $\lambda$ in terms of $\left(f_{0}, g_{0}\right)$, as well as $\mu$ in terms of $\left(\lambda, f_{0}, f_{1}, g_{1}\right)$.

  Deduce that the Bessel equation

  $$\frac{d^{2} u}{d z^{2}}+\frac{1}{z} \frac{d u}{d z}+\left(1-\frac{\nu^{2}}{z^{2}}\right) u=0$$

  where $\nu$ is a complex constant, has two solutions of the form

  $$\begin{aligned} &u^{(1)}(z)=\frac{e^{i z}}{z^{1 / 2}}\left[1+\frac{a^{(1)}}{z}+O\left(\frac{1}{z^{2}}\right)\right], \quad z \rightarrow \infty \\ &u^{(2)}(z)=\frac{e^{-i z}}{z^{1 / 2}}\left[1+\frac{a^{(2)}}{z}+O\left(\frac{1}{z^{2}}\right)\right], \quad z \rightarrow \infty \end{aligned}$$

  and determine $a^{(1)}$ and $a^{(2)}$ in terms of $\nu .$

  Can the above asymptotic expansions be valid for all $\arg (z)$, or are they valid only in certain domains of the complex $z$-plane? Justify your answer briefly.

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• Paper 3, Section II, $27 \mathrm{C}$

  #Asymptotic Methods

  2015

  Show that

  $$\int_{0}^{1} e^{i k t^{3}} d t=I_{1}-I_{2}, \quad k>0$$

  where $I_{1}$ is an integral from 0 to $\infty$ along the line $\arg (z)=\frac{\pi}{6}$ and $I_{2}$ is an integral from 1 to $\infty$ along a steepest-descent contour $C$ which you should determine.

  By employing in the integrals $I_{1}$ and $I_{2}$ the changes of variables $u=-i z^{3}$ and $u=-i\left(z^{3}-1\right)$, respectively, compute the first two terms of the large $k$ asymptotic expansion of the integral above.

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• Paper 1, Section II, C

  #Asymptotic Methods

  2015

  (a) State the integral expression for the gamma function $\Gamma(z)$, for $\operatorname{Re}(z)>0$, and express the integral

  $$\int_{0}^{\infty} t^{\gamma-1} e^{i t} d t, \quad 0<\gamma<1$$

  in terms of $\Gamma(\gamma)$. Explain why the constraints on $\gamma$ are necessary.

  (b) Show that

  $$\int_{0}^{\infty} \frac{e^{-k t^{2}}}{\left(t^{2}+t\right)^{\frac{1}{4}}} d t \sim \sum_{m=0}^{\infty} \frac{a_{m}}{k^{\alpha+\beta m}}, \quad k \rightarrow \infty$$

  for some constants $a_{m}, \alpha$ and $\beta$. Determine the constants $\alpha$ and $\beta$, and express $a_{m}$ in terms of the gamma function.

  State without proof the basic result needed for the rigorous justification of the above asymptotic formula.

  [You may use the identity:

  $$\left.(1+z)^{\alpha}=\sum_{m=0}^{\infty} c_{m} z^{m}, \quad c_{m}=\frac{\Gamma(\alpha+1)}{m ! \Gamma(\alpha+1-m)}, \quad|z|<1 .\right]$$

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• Paper 4, Section I, D

  #Classical Dynamics

  2015

  A triatomic molecule is modelled by three masses moving in a line while connected to each other by two identical springs of force constant $k$ as shown in the figure.

  

  (a) Write down the Lagrangian and derive the equations describing the motion of the atoms.

  (b) Find the normal modes and their frequencies. What motion does the lowest frequency represent?

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• Paper 3, Section I, $7 \mathrm{D}$

  #Classical Dynamics

  2015

  (a) Consider a particle of mass $m$ that undergoes periodic motion in a one-dimensional potential $V(q)$. Write down the Hamiltonian $H(p, q)$ for the system. Explain what is meant by the angle-action variables $(\theta, I)$ of the system and write down the integral expression for the action variable $I$.

  (b) For $V(q)=\frac{1}{2} m \omega^{2} q^{2}$ and fixed total energy $E$, describe the shape of the trajectories in phase-space. By using the expression for the area enclosed by the trajectory, or otherwise, find the action variable $I$ in terms of $\omega$ and $E$. Hence describe how $E$ changes with $\omega$ if $\omega$ varies slowly with time. Justify your answer.

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• Paper 2, Section I, D

  #Classical Dynamics

  2015

  The Lagrangian for a heavy symmetric top of mass $M$, pinned at a point that is a distance $l$ from the centre of mass, is

  $$L=\frac{1}{2} I_{1}\left(\dot{\theta}^{2}+\dot{\phi}^{2} \sin ^{2} \theta\right)+\frac{1}{2} I_{3}(\dot{\psi}+\dot{\phi} \cos \theta)^{2}-M g l \cos \theta$$

  (a) Find all conserved quantities. In particular, show that $\omega_{3}$, the spin of the top, is constant.

  (b) Show that $\theta$ obeys the equation of motion

  $$I_{1} \ddot{\theta}=-\frac{d V_{\text {eff }}}{d \theta},$$

  where the explicit form of $V_{\text {eff }}$ should be determined.

  (c) Determine the condition for uniform precession with no nutation, that is $\dot{\theta}=0$ and $\dot{\phi}=$ const. For what values of $\omega_{3}$ does such uniform precession occur?

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• Paper 1, Section I, D

  #Classical Dynamics

  2015

  (a) The action for a one-dimensional dynamical system with a generalized coordinate $q$ and Lagrangian $L$ is given by

  $$S=\int_{t_{1}}^{t_{2}} L(q, \dot{q}, t) d t$$

  State the principle of least action and derive the Euler-Lagrange equation.

  (b) A planar spring-pendulum consists of a light rod of length $l$ and a bead of mass $m$, which is able to slide along the rod without friction and is attached to the ends of the rod by two identical springs of force constant $k$ as shown in the figure. The rod is pivoted at one end and is free to swing in a vertical plane under the influence of gravity.

  

  (i) Identify suitable generalized coordinates and write down the Lagrangian of the system.

  (ii) Derive the equations of motion.

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• Paper 4, Section II, C

  #Classical Dynamics

  2015

  Consider a rigid body with angular velocity $\boldsymbol{\omega}$, angular momentum $\mathbf{L}$ and position vector $\mathbf{r}$, in its body frame.

  (a) Use the expression for the kinetic energy of the body,

  $$\frac{1}{2} \int d^{3} \mathbf{r} \rho(\mathbf{r}) \dot{\mathbf{r}}^{2},$$

  to derive an expression for the tensor of inertia of the body, I. Write down the relationship between $\mathbf{L}, \mathbf{I}$ and $\boldsymbol{\omega}$.

  (b) Euler's equations of torque-free motion of a rigid body are

  $$\begin{aligned} &I_{1} \dot{\omega}_{1}=\left(I_{2}-I_{3}\right) \omega_{2} \omega_{3} \\ &I_{2} \dot{\omega}_{2}=\left(I_{3}-I_{1}\right) \omega_{3} \omega_{1} \\ &I_{3} \dot{\omega}_{3}=\left(I_{1}-I_{2}\right) \omega_{1} \omega_{2} \end{aligned}$$

  Working in the frame of the principal axes of inertia, use Euler's equations to show that the energy $E$ and the squared angular momentum $\mathbf{L}^{2}$ are conserved.

  (c) Consider a cuboid with sides $a, b$ and $c$, and with mass $M$ distributed uniformly.

  (i) Use the expression for the tensor of inertia derived in (a) to calculate the principal moments of inertia of the body.

  (ii) Assume $b=2 a$ and $c=4 a$, and suppose that the initial conditions are such that

  $$\mathbf{L}^{2}=2 I_{2} E$$

  with the initial angular velocity $\omega$ perpendicular to the intermediate principal axis $\mathbf{e}_{2}$. Derive the first order differential equation for $\omega_{2}$ in terms of $E, M$ and $a$ and hence determine the long-term behaviour of $\boldsymbol{\omega}$.

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• Paper 2, Section II, C

  #Classical Dynamics

  2015

  (a) Consider a Lagrangian dynamical system with one degree of freedom. Write down the expression for the Hamiltonian of the system in terms of the generalized velocity $\dot{q}$, momentum $p$, and the Lagrangian $L(q, \dot{q}, t)$. By considering the differential of the Hamiltonian, or otherwise, derive Hamilton's equations.

  Show that if $q$ is ignorable (cyclic) with respect to the Lagrangian, i.e. $\partial L / \partial q=0$, then it is also ignorable with respect to the Hamiltonian.

  (b) A particle of charge $q$ and mass $m$ moves in the presence of electric and magnetic fields such that the scalar and vector potentials are $\phi=y E$ and $\mathbf{A}=(0, x B, 0)$, where $(x, y, z)$ are Cartesian coordinates and $E, B$ are constants. The Lagrangian of the particle is

  $$L=\frac{1}{2} m \dot{\mathbf{r}}^{2}-q \phi+q \dot{\mathbf{r}} \cdot \mathbf{A}$$

  Starting with the Lagrangian, derive an explicit expression for the Hamiltonian and use Hamilton's equations to determine the motion of the particle.

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• Paper 4, Section I, G

  #Coding and Cryptography

  2015

  Explain how to construct binary Reed-Muller codes. State and prove a result giving the minimum distance for each such Reed-Muller code.

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• Paper 3, Section I, G

  #Coding and Cryptography

  2015

  Let $A$ be a random variable that takes each value $a$ in the finite alphabet $\mathcal{A}$ with probability $p(a)$. Show that, if each $l(a)$ is an integer greater than 0 and $\sum 2^{-l(a)} \leqslant 1$, then there is a decodable binary code $c: \mathcal{A} \rightarrow\{0,1\}^{*}$ with each codeword $c(a)$ having length $l(a)$.

  Prove that, for any decodable code $c: \mathcal{A} \rightarrow\{0,1\}^{*}$, we have

  $$H(A) \leqslant \mathbb{E} l(A)$$

  where $H(A)$ is the entropy of the random variable $A$. When is there equality in this inequality?

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• Paper 2, Section I, G

  #Coding and Cryptography

  2015

  A random variable $A$ takes values in the alphabet $\mathcal{A}=\{a, b, c, d, e\}$ with probabilities $0.4,0.2,0.2,0.1$ and $0.1$. Calculate the entropy of $A$.

  Define what it means for a code for a general finite alphabet to be optimal. Find such a code for the distribution above and show that there are optimal codes for this distribution with differing lengths of codeword.

  [You may use any results from the course without proof. Note that $\log _{2} 5 \simeq 2.32$.]

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• Paper 1, Section I, $3 G$

  #Coding and Cryptography

  2015

  Let $\mathcal{A}$ be a finite alphabet. Explain what is meant by saying that a binary code $c: \mathcal{A} \rightarrow\{0,1\}^{*}$ has minimum distance $\delta$. If $c$ is such a binary code with minimum distance $\delta$, show that $c$ is $\delta-1$ error-detecting and $\left\lfloor\frac{1}{2}(\delta-1)\right\rfloor$ error-correcting.

  Show that it is possible to construct a code that has minimum distance $\delta$ for any integer $\delta>0$.

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• Paper 1, Section II, G

  #Coding and Cryptography

  2015

  Define the Hamming code. Show that it is a perfect, linear, 1-error correcting code.

  I wish to send a message through a noisy channel to a friend. The message consists of a large number $N=1,000$ of letters from a 16 -letter alphabet $\mathcal{A}$. When my friend has decoded the message, she can tell whether there have been any errors. If there have, she asks me to send the message again and this is repeated until she has received the message without error. For each individual binary digit that is transmitted, there is independently a small probability $p=0.001$ of an error.

  (a) Suppose that I encode my message by writing each letter as a 4-bit binary string. The whole message is then $4 N$ bits long. What is the probability $P$ that the entire message is transmitted without error? How many times should I expect to transmit the message until my friend receives it without error?

  (b) As an alternative, I use the Hamming code to encode each letter of $\mathcal{A}$ as a 7-bit binary string. What is the probability that my friend can decode a single 7-bit string correctly? Deduce that the probability $Q$ that the entire message is correctly decoded is given approximately by

  $$Q \simeq\left(1-21 p^{2}\right)^{N} \simeq \exp \left(-21 N p^{2}\right)$$

  Which coding method is better?

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• Paper 2, Section II, G

  #Coding and Cryptography

  2015

  Briefly describe the $R S A$ public key cipher.

  Just before it went into liquidation, the Internet Bank decided that it wanted to communicate with each of its customers using an RSA cipher. So, it chose a large modulus $N$, which is the product of two large prime numbers, and chose encrypting exponents $e_{j}$ and decrypting exponents $d_{j}$ for each customer $j$. The bank published $N$ and $e_{j}$ and sent the decrypting exponent $d_{j}$ secretly to customer $j$. Show explicitly that the cipher can be broken by each customer.

  The bank sent out the same message to each customer. I am not a customer of the bank but have two friends who are and I notice that their published encrypting exponents are coprime. Explain how I can find the original message. Can I break the cipher?

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• Paper 4, Section I, C

  #Cosmology

  2015

  Calculate the total effective number of relativistic spin states $g_{*}$ present in the early universe when the temperature $T$ is $10^{10} \mathrm{~K}$ if there are three species of low-mass neutrinos and antineutrinos in addition to photons, electrons and positrons. If the weak interaction rate is $\Gamma=\left(T / 10^{10} \mathrm{~K}\right)^{5} \mathrm{~s}^{-1}$ and the expansion rate of the universe is $H=\sqrt{8 \pi G \rho / 3}$, where $\rho$ is the total density of the universe, calculate the temperature $T_{*}$ at which the neutrons and protons cease to interact via weak interactions, and show that $T_{*} \propto g_{*}^{1 / 6}$.

  State the formula for the equilibrium ratio of neutrons to protons at $T_{*}$, and briefly describe the sequence of events as the temperature falls from $T_{*}$ to the temperature at which the nucleosynthesis of helium and deuterium ends.

  What is the effect of an increase or decrease of $g_{*}$ on the abundance of helium-4 resulting from nucleosynthesis? Why do changes in $g_{*}$ have a very small effect on the final abundance of deuterium?

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• Paper 3, Section I, C

  #Cosmology

  2015

  What is the flatness problem? Show by reference to the Friedmann equation how a period of accelerated expansion of the scale factor $a(t)$ in the early stages of the universe can solve the flatness problem if $\rho+3 P<0$, where $\rho$ is the mass density and $P$ is the pressure.

  In the very early universe, where we can neglect the spatial curvature and the cosmological constant, there is a homogeneous scalar field $\phi$ with a vacuum potential energy

  $$V(\phi)=m^{2} \phi^{2},$$

  and the Friedmann energy equation (in units where $8 \pi G=1$ ) is

  $$3 H^{2}=\frac{1}{2} \dot{\phi}^{2}+V(\phi),$$

  where $H$ is the Hubble parameter. The field $\phi$ obeys the evolution equation

  $$\ddot{\phi}+3 H \dot{\phi}+\frac{d V}{d \phi}=0$$

  During inflation, $\phi$ evolves slowly after starting from a large initial value $\phi_{i}$ at $t=0$. State what is meant by the slow-roll approximation. Show that in this approximation,

  $$\begin{aligned} \phi(t) &=\phi_{i}-\frac{2}{\sqrt{3}} m t \\ a(t) &=a_{i} \exp \left[\frac{m \phi_{i}}{\sqrt{3}} t-\frac{1}{3} m^{2} t^{2}\right]=a_{i} \exp \left[\frac{\phi_{i}^{2}-\phi^{2}(t)}{4}\right] \end{aligned}$$

  where $a_{i}$ is the initial value of $a$.

  As $\phi(t)$ decreases from its initial value $\phi_{i}$, what is its approximate value when the slow-roll approximation fails?

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• Paper 2, Section I, C

  #Cosmology

  2015

  The mass density perturbation equation for non-relativistic matter $\left(P \ll \rho c^{2}\right)$ with wave number $k$ in the late universe $\left(t>t_{\mathrm{eq}}\right)$ is

  $$\ddot{\delta}+2 \frac{\dot{a}}{a} \dot{\delta}-\left(4 \pi G \rho-\frac{c_{s}^{2} k^{2}}{a^{2}}\right) \delta=0 .$$

  Suppose that a non-relativistic fluid with the equation of state $P \propto \rho^{4 / 3}$ dominates the universe when $a(t)=t^{2 / 3}$, and the curvature and the cosmological constant can be neglected. Show that the sound speed can be written in the form $c_{s}^{2}(t) \equiv d P / d \rho=$ $\bar{c}_{s}^{2} t^{-2 / 3}$ where $\bar{c}_{s}$ is a constant.

  Find power-law solutions to $(*)$ of the form $\delta \propto t^{\beta}$ and hence show that the general solution is

  $$\delta=A_{k} t^{n_{+}}+B_{k} t^{n_{-}}$$

  where

  $$n_{\pm}=-\frac{1}{6} \pm\left[\left(\frac{5}{6}\right)^{2}-\bar{c}_{s}^{2} k^{2}\right]^{1 / 2}$$

  Interpret your solutions in the two regimes $k \ll k_{J}$ and $k \gg k_{J}$ where $k_{J}=\frac{5}{6 \bar{c}_{s}}$.

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• Paper 1, Section I, C

  #Cosmology

  2015

  Consider three galaxies $O, A$ and $B$ with position vectors $\mathbf{r}_{O}, \mathbf{r}_{A}$ and $\mathbf{r}_{B}$ in a homogeneous universe. Assuming they move with non-relativistic velocities $\mathbf{v}_{O}=\mathbf{0}, \mathbf{v}_{A}$ and $\mathbf{v}_{B}$, show that spatial homogeneity implies that the velocity field $\mathbf{v}(\mathbf{r})$ satisfies

  $$\mathbf{v}\left(\mathbf{r}_{B}-\mathbf{r}_{A}\right)=\mathbf{v}\left(\mathbf{r}_{B}-\mathbf{r}_{O}\right)-\mathbf{v}\left(\mathbf{r}_{A}-\mathbf{r}_{O}\right)$$

  and hence that $\mathbf{v}$ is linearly related to $\mathbf{r}$ by

  $$v_{i}=\sum_{j=1}^{3} H_{i j} r_{j}$$

  where the components of the matrix $H_{i j}$ are independent of $\mathbf{r}$.

  Suppose the matrix $H_{i j}$ has the form

  $$H_{i j}=\frac{D}{t}\left(\begin{array}{ccc} 5 & -1 & -2 \\ 1 & 5 & -1 \\ 2 & 1 & 5 \end{array}\right)$$

  with $D>0$ constant. Describe the kinematics of the cosmological expansion.

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• Paper 3, Section II, C

  #Cosmology

  2015

  Massive particles and antiparticles each with mass $m$ and respective number densities $n(t)$ and $\bar{n}(t)$ are present at time $t$ in the radiation era of an expanding universe with zero curvature and no cosmological constant. Assuming they interact with crosssection $\sigma$ at speed $v$, explain, by identifying the physical significance of each of the terms, why the evolution of $n(t)$ is described by

  $$\frac{d n}{d t}=-3 \frac{\dot{a}}{a} n-\langle\sigma v\rangle n \bar{n}+P(t)$$

  where the expansion scale factor of the universe is $a(t)$, and where the meaning of $P(t)$ should be briefly explained. Show that

  $$(n-\bar{n}) a^{3}=\text { constant }$$

  Assuming initial particle-antiparticle symmetry, show that

  $$\frac{d\left(n a^{3}\right)}{d t}=\langle\sigma v\rangle\left(n_{\mathrm{eq}}^{2}-n^{2}\right) a^{3}$$

  where $n_{\mathrm{eq}}$ is the equilibrium number density at temperature $T$.

  Let $Y=n / T^{3}$ and $x=m / T$. Show that

  $$\frac{d Y}{d x}=-\frac{\lambda}{x^{2}}\left(Y^{2}-Y_{\mathrm{eq}}^{2}\right)$$

  where $\lambda=m^{3}\langle\sigma v\rangle / H_{m}$ and $H_{m}$ is the Hubble expansion rate when $T=m$.

  When $x>x_{f} \simeq 10$, the number density $n$ can be assumed to be depleted only by annihilations. If $\lambda$ is constant, show that as $x \rightarrow \infty$ at late time, $Y$ approaches a constant value given by

  $$Y=\frac{x_{f}}{\lambda}$$

  Why do you expect weakly interacting particles to survive in greater numbers than strongly interacting particles?

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• Paper 1, Section II, C

  #Cosmology

  2015

  A closed universe contains black-body radiation, has a positive cosmological constant $\Lambda$, and is governed by the equation

  $$\frac{\dot{a}^{2}}{a^{2}}=\frac{\Gamma}{a^{4}}-\frac{1}{a^{2}}+\frac{\Lambda}{3},$$

  where $a(t)$ is the scale factor and $\Gamma$ is a positive constant. Using the substitution $y=a^{2}$ and the boundary condition $y(0)=0$, deduce the boundary condition for $\dot{y}(0)$ and show that

  $$\ddot{y}=\frac{4 \Lambda}{3} y-2$$

  and hence that

  $$a^{2}(t)=\frac{3}{2 \Lambda}\left[1-\cosh \left(\sqrt{\frac{4 \Lambda}{3}} t\right)+\lambda \sinh \left(\sqrt{\frac{4 \Lambda}{3}} t\right)\right]$$

  Express the constant $\lambda$ in terms of $\Lambda$ and $\Gamma$.

  Sketch the graphs of $a(t)$ for the cases $\lambda>1$ and $0<\lambda<1$.

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• Paper 4, Section II, G

  #Differential Geometry

  2015

  Let $\mathrm{U}(n)$ denote the set of $n \times n$ unitary complex matrices. Show that $\mathrm{U}(n)$ is a smooth (real) manifold, and find its dimension. [You may use any general results from the course provided they are stated correctly.] For $A$ any matrix in $\mathrm{U}(n)$ and $H$ an $n \times n$ complex matrix, determine when $H$ represents a tangent vector to $\mathrm{U}(n)$ at $A$.

  Consider the tangent spaces to $\mathrm{U}(n)$ equipped with the metric induced from the standard (Euclidean) inner product $\langle\cdot, \cdot\rangle$ on the real vector space of $n \times n$ complex matrices, given by $\langle L, K\rangle=\operatorname{Re} \operatorname{trace}\left(L K^{*}\right)$, where $\operatorname{Re}$ denotes the real part and $K^{*}$ denotes the conjugate transpose of $K$. Suppose that $H$ represents a tangent vector to $\mathrm{U}(n)$ at the identity matrix $I$. Sketch an explicit construction of a geodesic curve on $\mathrm{U}(n)$ passing through $I$ and with tangent direction $H$, giving a brief proof that the acceleration of the curve is always orthogonal to the tangent space to $\mathrm{U}(n)$.

  [Hint: You will find it easier to work directly with $n \times n$ complex matrices, rather than the corresponding $2 n \times 2 n$ real matrices.]

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• Paper 3 , Section II, G

  #Differential Geometry

  2015

  Show that the surface $S$ of revolution $x^{2}+y^{2}=\cosh ^{2} z$ in $\mathbb{R}^{3}$ is homeomorphic to a cylinder and has everywhere negative Gaussian curvature. Show moreover the existence of a closed geodesic on $S$.

  Let $S \subset \mathbb{R}^{3}$ be an arbitrary embedded surface which is homeomorphic to a cylinder and has everywhere negative Gaussian curvature. By using a suitable version of the Gauss-Bonnet theorem, show that $S$ contains at most one closed geodesic. [If required, appropriate forms of the Jordan curve theorem in the plane may also be used without proof.

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• Paper 2, Section II, G

  #Differential Geometry

  2015

  If $U$ denotes a domain in $\mathbb{R}^{2}$, what is meant by saying that a smooth map $\phi: U \rightarrow \mathbb{R}^{3}$ is an immersion? Define what it means for such an immersion to be isothermal. Explain what it means to say that an immersed surface is minimal.

  Let $\phi(u, v)=(x(u, v), y(u, v), z(u, v))$ be an isothermal immersion. Show that it is minimal if and only if $x, y, z$ are harmonic functions of $u, v$. [You may use the formula for the mean curvature given in terms of the first and second fundamental forms, namely $\left.H=(e G-2 f F+g E) /\left(2\left\{E G-F^{2}\right\}\right) .\right]$

  Produce an example of an immersed minimal surface which is not an open subset of a catenoid, helicoid, or a plane. Prove that your example does give an immersed minimal surface in $\mathbb{R}^{3}$.

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• Paper 1, Section II, $22 G$

  #Differential Geometry

  2015

  Let $\Omega \subset \mathbb{R}^{2}$ be a domain (connected open subset) with boundary $\partial \Omega$ a continuously differentiable simple closed curve. Denoting by $A(\Omega)$ the area of $\Omega$ and $l(\partial \Omega)$ the length of the curve $\partial \Omega$, state and prove the isoperimetric inequality relating $A(\Omega)$ and $l(\partial \Omega)$ with optimal constant, including the characterization for equality. [You may appeal to Wirtinger's inequality as long as you state it precisely.]

  Does the result continue to hold if the boundary $\partial \Omega$ is allowed finitely many points at which it is not differentiable? Briefly justify your answer by giving either a counterexample or an indication of a proof.

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• Paper 4, Section II, B

  #Dynamical Systems

  2015

  Let $f: I \rightarrow I$ be a continuous one-dimensional map of an interval $I \subset \mathbb{R}$. Explain what is meant by the statements (i) that $f$ has a horseshoe and (ii) that $f$ is chaotic (according to Glendinning's definition).

  Assume that $f$ has a 3-cycle $\left\{x_{0}, x_{1}, x_{2}\right\}$ with $x_{1}=f\left(x_{0}\right), x_{2}=f\left(x_{1}\right), x_{0}=f\left(x_{2}\right)$ and, without loss of generality, $x_{0}<x_{1}<x_{2}$. Prove that $f^{2}$ has a horseshoe. [You may assume the intermediate value theorem.]

  Represent the effect of $f$ on the intervals $I_{a}=\left[x_{0}, x_{1}\right]$ and $I_{b}=\left[x_{1}, x_{2}\right]$ by means of a directed graph, explaining carefully how the graph is constructed. Explain what feature of the graph implies the existence of a 3-cycle.

  The map $g: I \rightarrow I$ has a 5-cycle $\left\{x_{0}, x_{1}, x_{2}, x_{3}, x_{4}\right\}$ with $x_{i+1}=g\left(x_{i}\right), 0 \leqslant i \leqslant 3$ and $x_{0}=g\left(x_{4}\right)$, and $x_{0}<x_{1}<x_{2}<x_{3}<x_{4}$. For which $n, 1 \leqslant n \leqslant 4$, is an $n$-cycle of $g$ guaranteed to exist? Is $g$ guaranteed to be chaotic? Is $g$ guaranteed to have a horseshoe? Justify your answers. [You may use a suitable directed graph as part of your arguments.]

  How do your answers to the above change if instead $x_{4}<x_{2}<x_{1}<x_{3}<x_{0}$ ?

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• Paper 3, Section II, B

  #Dynamical Systems

  2015

  Consider the dynamical system

  $$\begin{aligned} &\dot{x}=-\mu+x^{2}-y \\ &\dot{y}=y(a-x) \end{aligned}$$

  where $a$ is to be regarded as a fixed real constant and $\mu$ as a real parameter.

  Find the fixed points of the system and determine the stability of the system linearized about the fixed points. Hence identify the values of $\mu$ at given $a$ where bifurcations occur.

  Describe informally the concepts of centre manifold theory and apply it to analyse the bifurcations that occur in the above system with $a=1$. In particular, for each bifurcation derive an equation for the dynamics on the extended centre manifold and hence classify the bifurcation.

  What can you say, without further detailed calculation, about the case $a=0$ ?

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• Paper 2, Section II, B

  #Dynamical Systems

  2015

  (a) An autonomous dynamical system $\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})$ in $\mathbb{R}^{2}$ has a periodic orbit $\mathbf{x}=\mathbf{X}(t)$ with period $T$. The linearized evolution of a small perturbation $\mathbf{x}=\mathbf{X}(t)+\boldsymbol{\eta}(t)$ is given by $\eta_{i}(t)=\Phi_{i j}(t) \eta_{j}(0)$. Obtain the differential equation and initial condition satisfied by the matrix $\Phi(t)$.

  Define the Floquet multipliers of the orbit. Explain why one of the multipliers is always unity and give a brief argument to show that the other is given by

  $$\exp \left(\int_{0}^{T} \nabla \cdot \mathbf{f}(\mathbf{X}(t)) d t\right)$$

  (b) Use the energy-balance method for nearly Hamiltonian systems to find leading-order approximations to the two limit cycles of the equation

  $$\ddot{x}+\epsilon\left(2 \dot{x}^{3}+2 x^{3}-4 x^{4} \dot{x}-\dot{x}\right)+x=0$$

  where $0<\epsilon \ll 1$.

  Determine the stability of each limit cycle, giving reasoning where necessary.

  [You may assume that $\int_{0}^{2 \pi} \cos ^{4} \theta d \theta=3 \pi / 4$ and $\int_{0}^{2 \pi} \cos ^{6} \theta d \theta=5 \pi / 8$.]

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• Paper 1, Section II, 28B

  #Dynamical Systems

  2015

  (a) What is a Lyapunov function?

  Consider the dynamical system for $\mathbf{x}(t)=(x(t), y(t))$ given by

  $$\begin{aligned} &\dot{x}=-x+y+x\left(x^{2}+y^{2}\right) \\ &\dot{y}=-y-2 x+y\left(x^{2}+y^{2}\right) \end{aligned}$$

  Prove that the origin is asymptotically stable (quoting carefully any standard results that you use).

  Show that the domain of attraction of the origin includes the region $x^{2}+y^{2}<r_{1}^{2}$ where the maximum possible value of $r_{1}$ is to be determined.

  Show also that there is a region $E=\left\{\mathbf{x} \mid x^{2}+y^{2}>r_{2}^{2}\right\}$ such that $\mathbf{x}(0) \in E$ implies that $|\mathbf{x}(t)|$ increases without bound. Explain your reasoning carefully. Find the smallest possible value of $r_{2}$.

  (b) Now consider the dynamical system

  $$\begin{aligned} \dot{x} &=x-y-x\left(x^{2}+y^{2}\right) \\ \dot{y} &=y+2 x-y\left(x^{2}+y^{2}\right) \end{aligned}$$

  Prove that this system has a periodic solution (again, quoting carefully any standard results that you use).

  Demonstrate that this periodic solution is unique.

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• Paper 4, Section II, A

  #Electrodynamics

  2015

  A point particle of charge $q$ has trajectory $y^{\mu}(\tau)$ in Minkowski space, where $\tau$ is its proper time. The resulting electromagnetic field is given by the Liénard-Wiechert 4-potential

  $$A^{\mu}(x)=-\frac{q \mu_{0} c}{4 \pi} \frac{u^{\mu}\left(\tau_{*}\right)}{R^{\nu}\left(\tau_{*}\right) u_{\nu}\left(\tau_{*}\right)}, \quad \text { where } \quad R^{\nu}=x^{\nu}-y^{\nu}(\tau) \quad \text { and } \quad u^{\mu}=d y^{\mu} / d \tau$$

  Write down the condition that determines the point $y^{\mu}\left(\tau_{*}\right)$ on the trajectory of the particle for a given value of $x^{\mu}$. Express this condition in terms of components, setting $x^{\mu}=(c t, \mathbf{x})$ and $y^{\mu}=\left(c t^{\prime}, \mathbf{y}\right)$, and define the retarded time $t_{r}$.

  Suppose that the 3 -velocity of the particle $\mathbf{v}\left(t^{\prime}\right)=\dot{\mathbf{y}}\left(t^{\prime}\right)=d \mathbf{y} / d t^{\prime}$ is small in size compared to $c$, and suppose also that $r=|\mathbf{x}| \gg|\mathbf{y}|$. Working to leading order in $1 / r$ and to first order in $\mathbf{v}$, show that

  $$\phi(x)=\frac{q \mu_{0} c}{4 \pi r}\left(c+\hat{\mathbf{r}} \cdot \mathbf{v}\left(t_{r}\right)\right), \quad \mathbf{A}(x)=\frac{q \mu_{0}}{4 \pi r} \mathbf{v}\left(t_{r}\right), \quad \text { where } \quad \hat{\mathbf{r}}=\mathbf{x} / r$$

  Now assume that $t_{r}$ can be replaced by $t_{-}=t-(r / c)$ in the expressions for $\phi$ and $\mathbf{A}$ above. Calculate the electric and magnetic fields to leading order in $1 / r$ and hence show that the Poynting vector is (in this approximation)

  $$\mathbf{N}(x)=\frac{q^{2} \mu_{0}}{(4 \pi)^{2} c} \frac{\hat{\mathbf{r}}}{r^{2}}\left|\hat{\mathbf{r}} \times \dot{\mathbf{v}}\left(t_{-}\right)\right|^{2}$$

  If the charge $q$ is performing simple harmonic motion $\mathbf{y}\left(t^{\prime}\right)=A \mathbf{n} \cos \omega t^{\prime}$, where $\mathbf{n}$ is a unit vector and $A \omega \ll c$, find the total energy radiated during one period of oscillation.

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• Paper 3, Section II, 34A

  #Electrodynamics

  2015

  (i) Consider the action

  $$S=-\frac{1}{4 \mu_{0} c} \int\left(F_{\mu \nu} F^{\mu \nu}+2 \lambda^{2} A_{\mu} A^{\mu}\right) d^{4} x+\frac{1}{c} \int A_{\mu} J^{\mu} d^{4} x$$

  where $A_{\mu}(x)$ is a 4-vector potential, $F_{\mu \nu}=\partial_{\mu} A_{\nu}-\partial_{\nu} A_{\mu}$ is the field strength tensor, $J^{\mu}(x)$ is a conserved current, and $\lambda \geqslant 0$ is a constant. Derive the field equation

  $$\partial_{\mu} F^{\mu \nu}-\lambda^{2} A^{\nu}=-\mu_{0} J^{\nu} .$$

  For $\lambda=0$ the action $S$ describes standard electromagnetism. Show that in this case the theory is invariant under gauge transformations of the field $A_{\mu}(x)$, which you should define. Is the theory invariant under these same gauge transformations when $\lambda>0$ ?

  Show that when $\lambda>0$ the field equation above implies

  $$\partial_{\mu} \partial^{\mu} A^{\nu}-\lambda^{2} A^{\nu}=-\mu_{0} J^{\nu}$$

  Under what circumstances does $(*)$ hold in the case $\lambda=0$ ?

  (ii) Now suppose that $A_{\mu}(x)$ and $J_{\mu}(x)$ obeying $(*)$ reduce to static 3 -vectors $\mathbf{A}(\mathbf{x})$ and $\mathbf{J}(\mathbf{x})$ in some inertial frame. Show that there is a solution

  $$\mathbf{A}(\mathbf{x})=-\mu_{0} \int G\left(\left|\mathbf{x}-\mathbf{x}^{\prime}\right|\right) \mathbf{J}\left(\mathbf{x}^{\prime}\right) d^{3} \mathbf{x}^{\prime}$$

  for a suitable Green's function $G(R)$ with $G(R) \rightarrow 0$ as $R \rightarrow \infty$. Determine $G(R)$ for any $\lambda \geqslant 0$. [Hint: You may find it helpful to consider first the case $\lambda=0$ and then the case $\lambda>0$, using the result $\nabla^{2}\left(\frac{1}{R} f(R)\right)=\nabla^{2}\left(\frac{1}{R}\right) f(R)+\frac{1}{R} f^{\prime \prime}(R)$, where $\left.R=\left|\mathbf{x}-\mathbf{x}^{\prime}\right| .\right]$

  If $\mathbf{J}(\mathbf{x})$ is zero outside some bounded region, comment on the effect of the value of $\lambda$ on the behaviour of $\mathbf{A}(\mathbf{x})$ when $|\mathbf{x}|$ is large. [No further detailed calculations are required.]

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• Paper 1, Section II, A

  #Electrodynamics

  2015

  Briefly explain how to interpret the components of the relativistic stress-energy tensor in terms of the density and flux of energy and momentum in some inertial frame.

  (i) The stress-energy tensor of the electromagnetic field is

  $$T_{\mathrm{em}}^{\mu \nu}=\frac{1}{\mu_{0}}\left(F^{\mu \alpha} F_{\alpha}^{\nu}-\frac{1}{4} \eta^{\mu \nu} F^{\alpha \beta} F_{\alpha \beta}\right)$$

  where $F_{\mu \nu}$ is the field strength, $\eta_{\mu \nu}$ is the Minkowski metric, and $\mu_{0}$ is the permeability of free space. Show that $\partial_{\mu} T_{\mathrm{em}}^{\mu \nu}=-F_{\mu}^{\nu} J^{\mu}$, where $J^{\mu}$ is the current 4-vector.

  [ Maxwell's equations are $\partial_{\mu} F^{\mu \nu}=-\mu_{0} J^{\nu}$ and $\partial_{\rho} F_{\mu \nu}+\partial_{\nu} F_{\rho \mu}+\partial_{\mu} F_{\nu \rho}=0 .$ ]

  (ii) A fluid consists of point particles of rest mass $m$ and charge $q$. The fluid can be regarded as a continuum, with 4 -velocity $u^{\mu}(x)$ depending on the position $x$ in spacetime. For each $x$ there is an inertial frame $S_{x}$ in which the fluid particles at $x$ are at rest. By considering components in $S_{x}$, show that the fluid has a current 4-vector field

  $$J^{\mu}=q n_{0} u^{\mu}$$

  and a stress-energy tensor

  $$T_{\text {fluid }}^{\mu \nu}=m n_{0} u^{\mu} u^{\nu},$$

  where $n_{0}(x)$ is the proper number density of particles (the number of particles per unit spatial volume in $S_{x}$ in a small region around $x$ ). Write down the Lorentz 4-force on a fluid particle at $x$. By considering the resulting 4 -acceleration of the fluid, show that the total stress-energy tensor is conserved, i.e.

  $$\partial_{\mu}\left(T_{\mathrm{em}}^{\mu \nu}+T_{\text {fluid }}^{\mu \nu}\right)=0 .$$

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• Paper 4, Section II, E

  #Fluid Dynamics II

  2015

  A stationary inviscid fluid of thickness $h$ and density $\rho$ is located below a free surface at $y=h$ and above a deep layer of inviscid fluid of the same density in $y<0$ flowing with uniform velocity $U>0$ in the $\mathbf{e}_{x}$ direction. The base velocity profile is thus

  $$u=U, y<0 ; \quad u=0,0<y<h$$

  while the free surface at $y=h$ is maintained flat by gravity.

  By considering small perturbations of the vortex sheet at $y=0$ of the form $\eta=\eta_{0} e^{i k x+\sigma t}, k>0$, calculate the dispersion relationship between $k$ and $\sigma$ in the irrotational limit. By explicitly deriving that

  $$\operatorname{Re}(\sigma)=\pm \frac{\sqrt{\tanh (h k)}}{1+\tanh (h k)} U k$$

  deduce that the vortex sheet is unstable at all wavelengths. Show that the growth rates of the unstable modes are approximately $U k / 2$ when $h k \gg 1$ and $U k \sqrt{h k}$ when $h k \ll 1$.

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• Paper 2, Section II, E

  #Fluid Dynamics II

  2015

  Consider an infinite rigid cylinder of radius a parallel to a horizontal rigid stationary surface. Let $\mathbf{e}_{x}$ be the direction along the surface perpendicular to the cylinder axis, $\mathbf{e}_{y}$ the direction normal to the surface (the surface is at $y=0$ ) and $\mathbf{e}_{z}$ the direction along the axis of the cylinder. The cylinder moves with constant velocity $U \mathbf{e}_{x}$. The minimum separation between the cylinder and the surface is denoted by $h_{0} \ll a$.

  (i) What are the conditions for the flow in the thin gap between the cylinder and the surface to be described by the lubrication equations? State carefully the relevant length scale in the $\mathbf{e}_{x}$ direction.

  (ii) Without doing any calculation, explain carefully why, in the lubrication limit, the net fluid force $\mathbf{F}$ acting on the stationary surface at $y=0$ has no component in the $\mathbf{e}_{y}$ direction.

  (iii) Using the lubrication approximation, calculate the $\mathbf{e}_{x}$ component of the velocity field in the gap between the cylinder and the surface, and determine the pressure gradient as a function of the gap thickness $h(x)$.

  (iv) Compute the tangential component of the force, $\mathbf{e}_{x} \cdot \mathbf{F}$, acting on the bottom surface per unit length in the $\mathbf{e}_{z}$ direction.

  [You may quote the following integrals:

  $$\int_{-\infty}^{\infty} \frac{d u}{\left(1+u^{2}\right)}=\pi, \quad \int_{-\infty}^{\infty} \frac{d u}{\left(1+u^{2}\right)^{2}}=\frac{\pi}{2}, \quad \int_{-\infty}^{\infty} \frac{d u}{\left(1+u^{2}\right)^{3}}=\frac{3 \pi}{8}$$

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• Paper 3, Section II, E

  #Fluid Dynamics II

  2015

  Consider a three-dimensional high-Reynolds number jet without swirl induced by a force $\mathbf{F}=F \mathbf{e}_{z}$ imposed at the origin in a fluid at rest. The velocity in the jet, described using cylindrical coordinates $(r, \theta, z)$, is assumed to remain steady and axisymmetric, and described by a boundary layer analysis.

  (i) Explain briefly why the flow in the jet can be described by the boundary layer equations

  $$u_{r} \frac{\partial u_{z}}{\partial r}+u_{z} \frac{\partial u_{z}}{\partial z}=\nu \frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial u_{z}}{\partial r}\right)$$

  (ii) Show that the momentum flux in the jet, $F=\int_{S} \rho u_{z}^{2} d S$, where $S$ is an infinite surface perpendicular to $\mathbf{e}_{z}$, is not a function of $z$. Combining this result with scalings from the boundary layer equations, derive the scalings for the unknown width $\delta(z)$ and typical velocity $U(z)$ of the jet as functions of $z$ and the other parameters of the problem $(\rho, \nu, F)$.

  (iii) Solving for the flow using a self-similar Stokes streamfunction

  $$\psi(r, z)=U(z) \delta^{2}(z) f(\eta), \quad \eta=r / \delta(z)$$

  show that $f(\eta)$ satisfies the differential equation

  $$f f^{\prime}-\eta\left(f^{\prime 2}+f f^{\prime \prime}\right)=f^{\prime}-\eta f^{\prime \prime}+\eta^{2} f^{\prime \prime \prime} .$$

  What boundary conditions should be applied to this equation? Give physical reasons for them.

  [Hint: In cylindrical coordinates for axisymmetric incompressible flow $\left(u_{r}(r, z), 0, u_{z}(r, z)\right)$ you are given the incompressibility condition as

  $$\frac{1}{r} \frac{\partial}{\partial r}\left(r u_{r}\right)+\frac{\partial u_{z}}{\partial z}=0$$

  the $z$-component of the Navier-Stokes equation as

  $$\rho\left(\frac{\partial u_{z}}{\partial t}+u_{r} \frac{\partial u_{z}}{\partial r}+u_{z} \frac{\partial u_{z}}{\partial z}\right)=-\frac{\partial p}{\partial z}+\mu\left[\frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial u_{z}}{\partial r}\right)+\frac{\partial^{2} u_{z}}{\partial z^{2}}\right]$$

  and the relationship between the Stokes streamfunction, $\psi(r, z)$, and the velocity components as

  $$\left.u_{r}=-\frac{1}{r} \frac{\partial \psi}{\partial z}, \quad u_{z}=\frac{1}{r} \frac{\partial \psi}{\partial r}\right]$$

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• Paper 1, Section II, E

  #Fluid Dynamics II

  2015

  (i) In a Newtonian fluid, the deviatoric stress tensor is linearly related to the velocity gradient so that the total stress tensor is

  $$\sigma_{i j}=-p \delta_{i j}+A_{i j k l} \frac{\partial u_{k}}{\partial x_{l}}$$

  Show that for an incompressible isotropic fluid with a symmetric stress tensor we necessarily have

  $$A_{i j k l} \frac{\partial u_{k}}{\partial x_{l}}=2 \mu e_{i j} \text {, }$$

  where $\mu$ is a constant which we call the dynamic viscosity and $e_{i j}$ is the symmetric part of $\partial u_{i} / \partial x_{j}$.

  (ii) Consider Stokes flow due to the translation of a rigid sphere $S_{a}$ of radius $a$ so that the sphere exerts a force $\mathbf{F}$ on the fluid. At distances much larger than the radius of the sphere, the instantaneous velocity and pressure fields are

  $$u_{i}(\mathbf{x})=\frac{1}{8 \mu \pi}\left(\frac{F_{i}}{r}+\frac{F_{m} x_{m} x_{i}}{r^{3}}\right), \quad p(\mathbf{x})=\frac{1}{4 \pi} \frac{F_{m} x_{m}}{r^{3}},$$

  where $\mathbf{x}$ is measured with respect to an origin located at the centre of the sphere, and $r=|\mathbf{x}|$.

  Consider a sphere $S_{R}$ of radius $R \gg a$ instantaneously concentric with $S_{a}$. By explicitly computing the tractions and integrating them, show that the force $G$ exerted by the fluid located in $r>R$ on $S_{R}$ is constant and independent of $R$, and evaluate it.

  (iii) Explain why the Stokes equations in the absence of body forces can be written

  $$\frac{\partial \sigma_{i j}}{\partial x_{j}}=0$$

  Show that by integrating this equation in the fluid volume located instantaneously between $S_{a}$ and $S_{R}$, you can recover the result in (ii) directly.

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• Paper 4, Section I, B

  #Further Complex Methods

  2015

  Explain how the Papperitz symbol

  $$P\left\{\begin{array}{cccc} z_{1} & z_{2} & z_{3} & \\ \alpha_{1} & \beta_{1} & \gamma_{1} & z \\ \alpha_{2} & \beta_{2} & \gamma_{2} & \end{array}\right\}$$

  represents a differential equation with certain properties. [You need not write down the differential equation explicitly.]

  The hypergeometric function $F(a, b, c ; z)$ is defined to be the solution of the equation given by the Papperitz symbol

  

  that is analytic at $z=0$ and such that $F(a, b, c ; 0)=1$. Show that

  $$F(a, b, c ; z)=(1-z)^{-a} F\left(a, c-b, c ; \frac{z}{z-1}\right) \text {, }$$

  indicating clearly any general results for manipulating Papperitz symbols that you use.

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• Paper 3, Section $I$, B

  #Further Complex Methods

  2015

  Define what is meant by the Cauchy principal value in the particular case

  $$\mathcal{P} \int_{-\infty}^{\infty} \frac{\cos x}{x^{2}-a^{2}} d x$$

  where the constant $a$ is real and strictly positive. Evaluate this expression explicitly, stating clearly any standard results involving contour integrals that you use.

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• Paper 2, Section I, B

  #Further Complex Methods

  2015

  Give a brief description of what is meant by analytic continuation.

  The dilogarithm function is defined by

  $$\mathrm{Li}_{2}(z)=\sum_{n=1}^{\infty} \frac{z^{n}}{n^{2}}, \quad|z|<1$$

  Let

  $$f(z)=-\int_{C} \frac{1}{u} \ln (1-u) d u$$

  where $C$ is a contour that runs from the origin to the point $z$. Show that $f(z)$ provides an analytic continuation of $\mathrm{Li}_{2}(z)$ and describe its domain of definition in the complex plane, given a suitable branch cut.

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• Paper 1, Section $I$, B

  #Further Complex Methods

  2015

  Evaluate the real integral

  $$\int_{0}^{\infty} \frac{x^{1 / 2} \ln x}{1+x^{2}} d x$$

  where $x^{1 / 2}$ is taken to be the positive square root.

  What is the value of

  $$\int_{0}^{\infty} \frac{x^{1 / 2}}{1+x^{2}} d x ?$$

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• Paper 2, Section II, B

  #Further Complex Methods

  2015

  The Riemann zeta function is defined by the sum

  $$\zeta(s)=\sum_{n=1}^{\infty} n^{-s}$$

  which converges for $\operatorname{Re} s>1$. Show that

  $$\zeta(s)=\frac{1}{\Gamma(s)} \int_{0}^{\infty} \frac{t^{s-1}}{e^{t}-1} d t, \quad \operatorname{Re} s>1$$

  The analytic continuation of $\zeta(s)$ is given by the Hankel contour integral

  $$\zeta(s)=\frac{\Gamma(1-s)}{2 \pi i} \int_{-\infty}^{0+} \frac{t^{s-1}}{e^{-t}-1} d t$$

  Verify that this agrees with the integral $(*)$ above when Re $s>1$ and $s$ is not an integer. [You may assume $\Gamma(s) \Gamma(1-s)=\pi / \sin \pi s$.] What happens when $s=2,3,4, \ldots$ ?

  Evaluate $\zeta(0)$. Show that $\left(e^{-t}-1\right)^{-1}+\frac{1}{2}$ is an odd function of $t$ and hence, or otherwise, show that $\zeta(-2 n)=0$ for any positive integer $n$.

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• Paper 1, Section II, B

  #Further Complex Methods

  2015

  Consider the differential equation

  $$x y^{\prime \prime}+(a-x) y^{\prime}-b y=0$$

  where $a$ and $b$ are constants with $\operatorname{Re}(b)>0$ and $\operatorname{Re}(a-b)>0$. Laplace's method for finding solutions involves writing

  $$y(x)=\int_{C} e^{x t} f(t) d t$$

  for some suitable contour $C$ and some suitable function $f(t)$. Determine $f(t)$ for the equation $(*)$ and use a clearly labelled diagram to specify contours $C$ giving two independent solutions when $x$ is real in each of the cases $x>0$ and $x<0$.

  Now let $a=3$ and $b=1$. Find explicit expressions for two independent solutions to $(*)$. Find, in addition, a solution $y(x)$ with $y(0)=1$.

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• Paper 3, Section II, F

  #Galois Theory

  2015

  Let $f \in \mathbb{Q}[t]$ be of degree $n>0$, with no repeated roots, and let $L$ be a splitting field for $f$.

  (i) Show that $f$ is irreducible if and only if for any $\alpha, \beta \in \operatorname{Root}_{f}(L)$ there is $\phi \in \operatorname{Gal}(L / \mathbb{Q})$ such that $\phi(\alpha)=\beta$.

  (ii) Explain how to define an injective homomorphism $\tau: \operatorname{Gal}(L / \mathbb{Q}) \rightarrow S_{n}$. Find an example in which the image of $\tau$ is the subgroup of $S_{3}$ generated by (2 3). Find another example in which $\tau$ is an isomorphism onto $S_{3}$.

  (iii) Let $f(t)=t^{5}-3$ and assume $f$ is irreducible. Find a chain of subgroups of $\operatorname{Gal}(L / \mathbb{Q})$ that shows it is a solvable group. [You may quote without proof any theorems from the course, provided you state them clearly.]

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• Paper 4, Section II, $17 \mathrm{~F}$

  #Galois Theory

  2015

  (i) Prove that a finite solvable extension $K \subseteq L$ of fields of characteristic zero is a radical extension.

  (ii) Let $x_{1}, \ldots, x_{7}$ be variables, $L=\mathbb{Q}\left(x_{1}, \ldots, x_{7}\right)$, and $K=\mathbb{Q}\left(e_{1}, \ldots, e_{7}\right)$ where $e_{i}$ are the elementary symmetric polynomials in the variables $x_{i}$. Is there an element $\alpha \in L$ such that $\alpha^{2} \in K$ but $\alpha \notin K$ ? Justify your answer.

  (iii) Find an example of a field extension $K \subseteq L$ of degree two such that $L \neq K(\sqrt{\alpha})$ for any $\alpha \in K$. Give an example of a field which has no extension containing an $11 t h$ primitive root of unity.

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• Paper 2, Section II, F

  #Galois Theory

  2015

  (i) State the fundamental theorem of Galois theory, without proof. Let $L$ be a splitting field of $t^{3}-2 \in \mathbb{Q}[t]$. Show that $\mathbb{Q} \subseteq L$ is Galois and that Gal $(L / \mathbb{Q})$ has a subgroup which is not normal.

  (ii) Let $\Phi_{8}$ be the 8 th cyclotomic polynomial and denote its image in $\mathbb{F}_{7}[t]$ again by $\Phi_{8}$. Show that $\Phi_{8}$ is not irreducible in $\mathbb{F}_{7}[t]$.

  (iii) Let $m$ and $n$ be coprime natural numbers, and let $\mu_{m}=\exp (2 \pi i / m)$ and $\mu_{n}=\exp (2 \pi i / n)$ where $i=\sqrt{-1}$. Show that $\mathbb{Q}\left(\mu_{m}\right) \cap \mathbb{Q}\left(\mu_{n}\right)=\mathbb{Q}$.

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• Paper 1, Section II, $17 \mathrm{~F}$

  #Galois Theory

  2015

  (i) Let $K \subseteq L$ be a field extension and $f \in K[t]$ be irreducible of positive degree. Prove the theorem which states that there is a $1-1$ correspondence

  $$\operatorname{Root}_{f}(L) \longleftrightarrow \operatorname{Hom}_{K}\left(\frac{K[t]}{\langle f\rangle}, L\right)$$

  (ii) Let $K$ be a field and $f \in K[t]$. What is a splitting field for $f$ ? What does it mean to say $f$ is separable? Show that every $f \in K[t]$ is separable if $K$ is a finite field.

  (iii) The primitive element theorem states that if $K \subseteq L$ is a finite separable field extension, then $L=K(\alpha)$ for some $\alpha \in L$. Give the proof of this theorem assuming $K$ is infinite.

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• Paper 4, Section II, D

  #General Relativity

  2015

  In static spherically symmetric coordinates, the metric $g_{a b}$ for de Sitter space is given by

  $$d s^{2}=-\left(1-r^{2} / a^{2}\right) d t^{2}+\left(1-r^{2} / a^{2}\right)^{-1} d r^{2}+r^{2} d \Omega^{2}$$

  where $d \Omega^{2}=d \theta^{2}+\sin ^{2} \theta d \phi^{2}$ and $a$ is a constant.

  (a) Let $u=t-a \tanh ^{-1}(r / a)$ for $r \leqslant a$. Use the $(u, r, \theta, \phi)$ coordinates to show that the surface $r=a$ is non-singular. Is $r=0$ a space-time singularity?

  (b) Show that the vector field $g^{a b} u_{, a}$ is null.

  (c) Show that the radial null geodesics must obey either

  $$\frac{d u}{d r}=0 \quad \text { or } \quad \frac{d u}{d r}=-\frac{2}{1-r^{2} / a^{2}}$$

  Which of these families of geodesics is outgoing $(d r / d t>0) ?$

  Sketch these geodesics in the $(u, r)$ plane for $0 \leqslant r \leqslant a$, where the $r$-axis is horizontal and lines of constant $u$ are inclined at $45^{\circ}$ to the horizontal.

  (d) Show, by giving an explicit example, that an observer moving on a timelike geodesic starting at $r=0$ can cross the surface $r=a$ within a finite proper time.

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• Paper 2, Section II, D

  #General Relativity

  2015

  (a) The Schwarzschild metric is

  $$d s^{2}=-\left(1-r_{s} / r\right) d t^{2}+\left(1-r_{s} / r\right)^{-1} d r^{2}+r^{2}\left(d \theta^{2}+\sin ^{2} \theta d \phi^{2}\right)$$

  (in units for which the speed of light $c=1$ ). Show that a timelike geodesic in the equatorial plane obeys

  $$\frac{1}{2} \dot{r}^{2}+V(r)=\frac{1}{2} E^{2}$$

  where

  $$2 V(r)=\left(1-\frac{r_{s}}{r}\right)\left(1+\frac{h^{2}}{r^{2}}\right)$$

  and $E$ and $h$ are constants.

  (b) For a circular orbit of radius $r$, show that

  $$h^{2}=\frac{r^{2} r_{s}}{2 r-3 r_{s}} .$$

  Given that the orbit is stable, show that $r>3 r_{s}$.

  (c) Alice lives on a small planet that is in a stable circular orbit of radius $r$ around a (non-rotating) black hole of radius $r_{s}$. Bob lives on a spacecraft in deep space far from the black hole and at rest relative to it. Bob is ageing $k$ times faster than Alice. Find an expression for $k^{2}$ in terms of $r$ and $r_{s}$ and show that $k<\sqrt{2}$.

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• Paper 3, Section II, D

  #General Relativity

  2015

  Let $\Gamma_{b c}^{a}$ be the Levi-Civita connection and $R_{b c d}^{a}$ the Riemann tensor corresponding to a metric $g_{a b}(x)$, and let $\widetilde{\Gamma}_{b c}^{a}$ be the Levi-Civita connection and $\widetilde{R}_{b c d}^{a}$ the Riemann tensor corresponding to a metric $\tilde{g}_{a b}(x)$. Let $T_{b c}^{a}=\widetilde{\Gamma}_{b c}^{a}-\Gamma_{b c}^{a}$.

  (a) Show that $T_{b c}^{a}$ is a tensor.

  (b) Using local inertial coordinates for the metric $g_{a b}$, or otherwise, show that

  $$\widetilde{R}_{b c d}^{a}-R_{b c d}^{a}=2 T_{b[d ; c]}^{a}-2 T_{e[d}^{a} T_{c] b}^{e}$$

  holds in all coordinate systems, where the semi-colon denotes covariant differentiation using the connection $\Gamma_{b c}^{a}$. [You may assume that $R_{b c d}^{a}=2 \Gamma_{b[d, c]}^{a}-2 \Gamma_{e[d}^{a} \Gamma_{c] b}^{e}$.]

  (c) In the case that $T_{b c}^{a}=\ell^{a} g_{b c}$ for some vector field $\ell^{a}$, show that $\widetilde{R}_{b d}=R_{b d}$ if and only if

  $$\ell_{b ; d}+\ell_{b} \ell_{d}=0$$

  (d) Using the result that $\ell_{[a ; b]}=0$ if and only if $\ell_{a}=\phi_{, a}$ for some scalar field $\phi$, show that the condition on $\ell_{a}$ in part (c) can be written as

  $$k_{a ; b}=0$$

  for a certain covector field $k_{a}$, which you should define.

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• Paper 1, Section II, 35D

  #General Relativity

  2015

  A vector field $\xi^{a}$ is said to be a conformal Killing vector field of the metric $g_{a b}$ if

  $$\xi_{(a ; b)}=\frac{1}{2} \psi g_{a b}$$

  for some scalar field $\psi$. It is a Killing vector field if $\psi=0$.

  (a) Show that $(*)$ is equivalent to

  $$\xi^{c} g_{a b, c}+\xi_{, a}^{c} g_{b c}+\xi_{, b}^{c} g_{a c}=\psi g_{a b}$$

  (b) Show that if $\xi^{a}$ is a conformal Killing vector field of the metric $g_{a b}$, then $\xi^{a}$ is a Killing vector field of the metric $\mathrm{e}^{2 \phi} g_{a b}$, where $\phi$ is any function that obeys

  $$2 \xi^{c} \phi_{, c}+\psi=0 .$$

  (c) Use part (b) to find an example of a metric with coordinates $t, x, y$ and $z$ (where $t>0)$ for which $(t, x, y, z)$ are the contravariant components of a Killing vector field. [Hint: You may wish to start by considering what happens in Minkowski space.]

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• Paper 1, Section II, I

  #Graph Theory

  2015

  (a) What does it mean to say that a graph $G$ is strongly regular with parameters $(k, a, b) ?$

  (b) Let $G$ be an incomplete, strongly regular graph with parameters $(k, a, b)$ and of order $n$. Suppose $b \geqslant 1$. Show that the numbers

  $$\frac{1}{2}\left(n-1 \pm \frac{(n-1)(b-a)-2 k}{\sqrt{(a-b)^{2}+4(k-b)}}\right)$$

  are integers.

  (c) Suppose now that $G$ is an incomplete, strongly regular graph with parameters $(k, 0,3)$. Show that $|G| \in\{6,162\}$.

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• Paper 2, Section II, I

  #Graph Theory

  2015

  (a) Define the Ramsey numbers $R(s, t)$ and $R(s)$ for integers $s, t \geqslant 2$. Show that $R(s, t)$ exists for all $s, t \geqslant 2$ and that if $s, t \geqslant 3$ then $R(s, t) \leqslant R(s-1, t)+R(s, t-1)$.

  (b) Show that, as $s \rightarrow \infty$, we have $R(s)=O\left(4^{s}\right)$ and $R(s)=\Omega\left(2^{s / 2}\right)$.

  (c) Show that, as $t \rightarrow \infty$, we have $R(3, t)=O\left(t^{2}\right)$ and $R(3, t)=\Omega\left(\left(\frac{t}{\log t}\right)^{3 / 2}\right)$.

  [Hint: For the lower bound in (c), you may wish to begin by modifying a random graph to show that for all $n$ and $p$ we have

  $$R(3, t)>n-\left(\begin{array}{l} n \\ 3 \end{array}\right) p^{3}-\left(\begin{array}{c} n \\ t \end{array}\right)(1-p)\left(\begin{array}{c} t \\ 2 \end{array}\right)$$

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• Paper 3, Section II, I

  #Graph Theory

  2015

  (a) Let $G$ be a graph. What is a Hamilton cycle in $G$ ? What does it mean to say that $G$ is Hamiltonian?

  (b) Let $G$ be a graph of order $n \geqslant 3$ satisfying $\delta(G) \geqslant \frac{n}{2}$. Show that $G$ is Hamiltonian. For each $n \geqslant 3$, exhibit a non-Hamiltonian graph $G_{n}$ of order $n$ with $\delta\left(G_{n}\right)=\left\lceil\frac{n}{2}\right\rceil-1$.

  (c) Let $H$ be a bipartite graph with $n \geqslant 2$ vertices in each class satisfying $\delta(H)>\frac{n}{2}$. Show that $H$ is Hamiltonian. For each $n \geqslant 2$, exhibit a non-Hamiltonian bipartite graph $H_{n}$ with $n$ vertices in each class and $\delta\left(H_{n}\right)=\left\lfloor\frac{n}{2}\right\rfloor$.

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• Paper 4, Section II, I

  #Graph Theory

  2015

  Let $G$ be a bipartite graph with vertex classes $X$ and $Y$. What does it mean to say that $G$ contains a matching from $X$ to $Y$ ? State and prove Hall's Marriage Theorem.

  Suppose now that every $x \in X$ has $d(x) \geqslant 1$, and that if $x \in X$ and $y \in Y$ with $x y \in E(G)$ then $d(x) \geqslant d(y)$. Show that $G$ contains a matching from $X$ to $Y$.

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• Paper 1, Section II, D

  #Integrable Systems

  2015

  Let $u_{t}=K\left(x, u, u_{x}, \ldots\right)$ be an evolution equation for the function $u=u(x, t)$. Assume $u$ and all its derivatives decay rapidly as $|x| \rightarrow \infty$. What does it mean to say that the evolution equation for $u$ can be written in Hamiltonian form?

  The modified KdV (mKdV) equation for $u$ is

  $$u_{t}+u_{x x x}-6 u^{2} u_{x}=0 .$$

  Show that small amplitude solutions to this equation are dispersive.

  Demonstrate that the mKdV equation can be written in Hamiltonian form and define the associated Poisson bracket $\{,$,} on the space of functionals of u. Verify that the Poisson bracket is linear in each argument and anti-symmetric.

  Show that a functional $I=I[u]$ is a first integral of the mKdV equation if and only if $\{I, H\}=0$, where $H=H[u]$ is the Hamiltonian.

  Show that if $u$ satisfies the mKdV equation then

  $$\frac{\partial}{\partial t}\left(u^{2}\right)+\frac{\partial}{\partial x}\left(2 u u_{x x}-u_{x}^{2}-3 u^{4}\right)=0$$

  Using this equation, show that the functional

  $$I[u]=\int u^{2} d x$$

  Poisson-commutes with the Hamiltonian.

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• Paper 2, Section II, D

  #Integrable Systems

  2015

  (a) Explain how a vector field

  $$V=\xi(x, u) \frac{\partial}{\partial x}+\eta(x, u) \frac{\partial}{\partial u}$$

  generates a 1-parameter group of transformations $g^{\epsilon}:(x, u) \mapsto(\tilde{x}, \tilde{u})$ in terms of the solution to an appropriate differential equation. [You may assume the solution to the relevant equation exists and is unique.]

  (b) Suppose now that $u=u(x)$. Define what is meant by a Lie-point symmetry of the ordinary differential equation

  $$\Delta\left[x, u, u^{(1)}, \ldots, u^{(n)}\right]=0, \quad \text { where } \quad u^{(k)} \equiv \frac{d^{k} u}{d x^{k}}, \quad k=1, \ldots, n$$

  (c) Prove that every homogeneous, linear ordinary differential equation for $u=u(x)$ admits a Lie-point symmetry generated by the vector field

  $$V=u \frac{\partial}{\partial u}$$

  By introducing new coordinates

  $$s=s(x, u), \quad t=t(x, u)$$

  which satisfy $V(s)=1$ and $V(t)=0$, show that every differential equation of the form

  $$\frac{d^{2} u}{d x^{2}}+p(x) \frac{d u}{d x}+q(x) u=0$$

  can be reduced to a first-order differential equation for an appropriate function.

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• Paper 3, Section II, D

  #Integrable Systems

  2015

  Let $L=L(t)$ and $A=A(t)$ be real $N \times N$ matrices, with $L$ symmetric and $A$ antisymmetric. Suppose that

  $$\frac{d L}{d t}=L A-A L$$

  Show that all eigenvalues of the matrix $L(t)$ are $t$-independent. Deduce that the coefficients of the polynomial

  $$P(x)=\operatorname{det}(x I-L(t))$$

  are first integrals of the system.

  What does it mean for a $2 n$-dimensional Hamiltonian system to be integrable? Consider the Toda system with coordinates $\left(q_{1}, q_{2}, q_{3}\right)$ obeying

  $$\frac{d^{2} q_{i}}{d t^{2}}=\mathrm{e}^{q_{i-1}-q_{i}}-\mathrm{e}^{q_{i}-q_{i+1}}, \quad i=1,2,3$$

  where here and throughout the subscripts are to be determined modulo 3 so that $q_{4} \equiv q_{1}$ and $q_{0} \equiv q_{3}$. Show that

  $$H\left(q_{i}, p_{i}\right)=\frac{1}{2} \sum_{i=1}^{3} p_{i}^{2}+\sum_{i=1}^{3} \mathrm{e}^{q_{i}-q_{i+1}}$$

  is a Hamiltonian for the Toda system.

  Set $a_{i}=\frac{1}{2} \exp \left(\frac{q_{i}-q_{i+1}}{2}\right)$ and $b_{i}=-\frac{1}{2} p_{i}$. Show that

  $$\frac{d a_{i}}{d t}=\left(b_{i+1}-b_{i}\right) a_{i}, \quad \frac{d b_{i}}{d t}=2\left(a_{i}^{2}-a_{i-1}^{2}\right), \quad i=1,2,3$$

  Is this coordinate transformation canonical?

  By considering the matrices

  $$L=\left(\begin{array}{lll} b_{1} & a_{1} & a_{3} \\ a_{1} & b_{2} & a_{2} \\ a_{3} & a_{2} & b_{3} \end{array}\right), \quad A=\left(\begin{array}{ccc} 0 & -a_{1} & a_{3} \\ a_{1} & 0 & -a_{2} \\ -a_{3} & a_{2} & 0 \end{array}\right)$$

  or otherwise, compute three independent first integrals of the Toda system. [Proof of independence is not required.]

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• Paper 3, Section II, G

  #Linear Analysis

  2015

  State and prove the Baire Category Theorem. [Choose any version you like.]

  An isometry from a metric space $(M, d)$ to another metric space $(N, e)$ is a function $\varphi: M \rightarrow N$ such that $e(\varphi(x), \varphi(y))=d(x, y)$ for all $x, y \in M$. Prove that there exists no isometry from the Euclidean plane $\ell_{2}^{2}$ to the Banach space $c_{0}$ of sequences converging to 0 . [Hint: Assume $\varphi: \ell_{2}^{2} \rightarrow \mathrm{c}_{0}$ is an isometry. For $n \in \mathbb{N}$ and $x \in \ell_{2}^{2}$ let $\varphi_{n}(x)$ denote the $n^{\text {th }}$ coordinate of $\varphi(x)$. Consider the sets $F_{n}$ consisting of all pairs $(x, y)$ with $\|x\|_{2}=\|y\|_{2}=1$ and $\|\varphi(x)-\varphi(y)\|_{\infty}=\left|\varphi_{n}(x)-\varphi_{n}(y)\right|$.]

  Show that for each $n \in \mathbb{N}$ there is a linear isometry $\ell_{1}^{n} \rightarrow \mathrm{c}_{0}$.

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• Paper 4, Section II, G

  #Linear Analysis

  2015

  Let $H$ be a Hilbert space and $T \in \mathcal{B}(H)$. Define what is meant by an adjoint of $T$ and prove that it exists, it is linear and bounded, and that it is unique. [You may use the Riesz Representation Theorem without proof.]

  What does it mean to say that $T$ is a normal operator? Give an example of a bounded linear map on $\ell_{2}$ that is not normal.

  Show that $T$ is normal if and only if $\|T x\|=\left\|T^{*} x\right\|$ for all $x \in H$.

  Prove that if $T$ is normal, then $\sigma(T)=\sigma_{\mathrm{ap}}(T)$, that is, that every element of the spectrum of $T$ is an approximate eigenvalue of $T$.

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• Paper 2, Section II, G

  #Linear Analysis

  2015

  (a) Let $T: X \rightarrow Y$ be a linear map between normed spaces. What does it mean to say that $T$ is bounded? Show that $T$ is bounded if and only if $T$ is continuous. Define the operator norm of $T$ and show that the set $\mathcal{B}(X, Y)$ of all bounded, linear maps from $X$ to $Y$ is a normed space in the operator norm.

  (b) For each of the following linear maps $T$, determine if $T$ is bounded. When $T$ is bounded, compute its operator norm and establish whether $T$ is compact. Justify your answers. Here $\|f\|_{\infty}=\sup _{t \in[0,1]}|f(t)|$ for $f \in C[0,1]$ and $\|f\|=\|f\|_{\infty}+\left\|f^{\prime}\right\|_{\infty}$ for $f \in C^{1}[0,1]$.

  (i) $T:\left(C^{1}[0,1],\|\cdot\|_{\infty}\right) \rightarrow\left(C^{1}[0,1],\|\cdot\|\right), T(f)=f$.

  (ii) $T:\left(C^{1}[0,1],\|\cdot\|\right) \rightarrow\left(C[0,1],\|\cdot\|_{\infty}\right), T(f)=f$.

  (iii) $T:\left(C^{1}[0,1],\|\cdot\|\right) \rightarrow\left(C[0,1],\|\cdot\|_{\infty}\right), T(f)=f^{\prime}$.

  (iv) $T:\left(C[0,1],\|\cdot\|_{\infty}\right) \rightarrow \mathbb{R}, T(f)=\int_{0}^{1} f(t) h(t) d t$, where $h$ is a given element of $C[0,1]$. [Hint: Consider first the case that $h(x) \neq 0$ for every $x \in[0,1]$, and apply $T$ to a suitable function. In the general case apply $T$ to a suitable sequence of functions.]

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• Paper 1, Section II, G

  #Linear Analysis

  2015

  (a) Let $\left(e_{n}\right)_{n=1}^{\infty}$ be an orthonormal basis of an inner product space $X$. Show that for all $x \in X$ there is a unique sequence $\left(a_{n}\right)_{n=1}^{\infty}$ of scalars such that $x=\sum_{n=1}^{\infty} a_{n} e_{n}$.

  Assume now that $X$ is a Hilbert space and that $\left(f_{n}\right)_{n=1}^{\infty}$ is another orthonormal basis of $X$. Prove that there is a unique bounded linear map $U: X \rightarrow X$ such that $U\left(e_{n}\right)=f_{n}$ for all $n \in \mathbb{N}$. Prove that this map $U$ is unitary.

  (b) Let $1 \leqslant p<\infty$ with $p \neq 2$. Show that no subspace of $\ell_{2}$ is isomorphic to $\ell_{p}$. [Hint: Apply the generalized parallelogram law to suitable vectors.]

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• Paper 4, Section II, I

  #Logic and Set Theory

  2015

  State the Axiom of Foundation and the Principle of $\epsilon$-Induction, and show that they are equivalent (in the presence of the other axioms of $Z F$ ). [You may assume the existence of transitive closures.]

  Explain briefly how the Principle of $\epsilon$-Induction implies that every set is a member of some $V_{\alpha}$.

  Find the ranks of the following sets:

  (i) $\{\omega+1, \omega+2, \omega+3\}$,

  (ii) the Cartesian product $\omega \times \omega$,

  (iii) the set of all functions from $\omega$ to $\omega^{2}$.

  [You may assume standard properties of rank.]

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• Paper 3, Section II, I

  #Logic and Set Theory

  2015

  (i) State and prove Zorn's Lemma. [You may assume Hartogs' Lemma.] Where in your proof have you made use of the Axiom of Choice?

  (ii) Let $<$ be a partial ordering on a set $X$. Prove carefully that $<$ may be extended to a total ordering of $X$.

  What does it mean to say that $<$ is well-founded?

  If $<$ has an extension that is a well-ordering, must $<$ be well-founded? If $<$ is well-founded, must every total ordering extending it be a well-ordering? Justify your answers.

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• Paper 2, Section II, I

  #Logic and Set Theory

  2015

  (a) Give the inductive and synthetic definitions of ordinal addition, and prove that they are equivalent. Give the inductive definitions of ordinal multiplication and ordinal exponentiation.

  (b) Answer, with brief justification, the following:

  (i) For ordinals $\alpha, \beta$ and $\gamma$ with $\alpha<\beta$, must we have $\alpha+\gamma<\beta+\gamma$ ? Must we have $\gamma+\alpha<\gamma+\beta$ ?

  (ii) For ordinals $\alpha$ and $\beta$ with $\alpha<\beta$, must we have $\alpha^{\omega}<\beta^{\omega}$ ?

  (iii) Is there an ordinal $\alpha>1$ such that $\alpha^{\omega}=\alpha$ ?

  (iv) Show that $\omega^{\omega_{1}}=\omega_{1}$. Is $\omega_{1}$ the least ordinal $\alpha$ such that $\omega^{\alpha}=\alpha$ ?

  [You may use standard facts about ordinal arithmetic.]

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• Paper 1, Section II, I

  #Logic and Set Theory

  2015

  State and prove the Completeness Theorem for Propositional Logic.

  [You do not need to give definitions of the various terms involved. You may assume the Deduction Theorem, provided that you state it precisely.]

  State the Compactness Theorem and the Decidability Theorem, and deduce them from the Completeness Theorem.

  Let $S$ consist of the propositions $p_{n+1} \Rightarrow p_{n}$ for $n=1,2,3, \ldots$. Does $S$ prove $p_{1}$ ? Justify your answer. [Here $p_{1}, p_{2}, p_{3}, \ldots$ are primitive propositions.]

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• Paper 4, Section I, E

  #Mathematical Biology

  2015

  (i) A variant of the classic logistic population model is given by the HutchinsonWright equation

  $$\frac{d x(t)}{d t}=\alpha x(t)[1-x(t-T)]$$

  where $\alpha, T>0$. Determine the condition on $\alpha$ (in terms of $T$ ) for the constant solution $x(t)=1$ to be stable.

  (ii) Another variant of the logistic model is given by the equation

  $$\frac{d x(t)}{d t}=\alpha\left[x(t-T)-x(t)^{2}\right]$$

  where $\alpha, T>0$. Give a brief interpretation of what this model represents.

  Determine the condition on $\alpha$ (in terms of $T$ ) for the constant solution $x(t)=1$ to be stable in this model.

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• Paper 3, Section I, E

  #Mathematical Biology

  2015

  The number of a certain type of annual plant in year $n$ is given by $x_{n}$. Each plant produces $k$ seeds that year and then dies before the next year. The proportion of seeds that germinate to produce a new plant the next year is given by $e^{-\gamma x_{n}}$ where $\gamma>0$. Explain briefly why the system can be described by

  $$x_{n+1}=k x_{n} e^{-\gamma x_{n}}$$

  Give conditions on $k$ for a stable positive equilibrium of the plant population size to be possible.

  Winters become milder and now a proportion $s$ of all plants survive each year $(s \in(0,1))$. Assume that plants produce seeds as before while they are alive. Show that a wider range of $k$ now gives a stable positive equilibrium population.

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• Paper 2, Section I, E

  #Mathematical Biology

  2015

  An activator-inhibitor system is described by the equations

  $$\begin{aligned} \frac{\partial u}{\partial t} &=2 u+u^{2}-u v+\frac{\partial^{2} u}{\partial x^{2}} \\ \frac{\partial v}{\partial t} &=a\left(u^{2}-v\right)+d \frac{\partial^{2} v}{\partial x^{2}} \end{aligned}$$

  where $a, d>0$.

  Find the range of $a$ for which the spatially homogeneous system has a stable equilibrium solution with $u>0$ and $v>0$.

  For the case when the homogeneous system is stable, consider spatial perturbations proportional to $\cos (k x)$ to the equilibrium solution found above. Show that the system has a Turing instability when

  $$d>\left(\frac{7}{2}+2 \sqrt{3}\right) a$$

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• Paper 1, Section I, E

  #Mathematical Biology

  2015

  The population density $n(a, t)$ of individuals of age $a$ at time $t$ satisfies

  $$\frac{\partial n}{\partial t}+\frac{\partial n}{\partial a}=-\mu(a) n(a, t), \quad n(0, t)=\int_{0}^{\infty} b(a) n(a, t) d a$$

  where $\mu(a)$ is the age-dependent death rate and $b(a)$ is the birth rate per individual of age $a$. Show that this may be solved with a similarity solution of the form $n(a, t)=e^{\gamma t} r(a)$ if the growth rate $\gamma$ satisfies $\phi(\gamma)=1$ where

  $$\phi(\gamma)=\int_{0}^{\infty} b(a) e^{-\gamma a-\int_{0}^{a} \mu(s) d s} d a$$

  Suppose now that the birth rate is given by $b(a)=B a^{p} e^{-\lambda a}$ with $B, \lambda>0$ and $p$ is a positive integer, and the death rate is constant in age (i.e. $\mu(a)=\mu)$. Find the average number of offspring per individual.

  Find the similarity solution, and find the threshold $B^{*}$ for the birth parameter $B$ so that $B>B^{*}$ corresponds to a growing population.

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• Paper 4, Section II, E

  #Mathematical Biology

  2015

  In a stochastic model of multiple populations, $P=P(\mathbf{x}, t)$ is the probability that the population sizes are given by the vector $\mathbf{x}$ at time $t$. The jump rate $W(\mathbf{x}, \mathbf{r})$ is the probability per unit time that the population sizes jump from $\mathbf{x}$ to $\mathbf{x}+\mathbf{r}$. Under suitable assumptions, the system may be approximated by the multivariate Fokker-Planck equation (with summation convention)

  $$\frac{\partial}{\partial t} P=-\frac{\partial}{\partial x_{i}} A_{i} P+\frac{1}{2} \frac{\partial^{2}}{\partial x_{i} \partial x_{j}} B_{i j} P$$

  where $A_{i}(\mathbf{x})=\sum_{\mathbf{r}} r_{i} W(\mathbf{x}, \mathbf{r})$ and matrix elements $B_{i j}(\mathbf{x})=\sum_{\mathbf{r}} r_{i} r_{j} W(\mathbf{x}, \mathbf{r})$.

  (a) Use the multivariate Fokker-Planck equation to show that

  $$\begin{aligned} \frac{d}{d t}\left\langle x_{k}\right\rangle &=\left\langle A_{k}\right\rangle \\ \frac{d}{d t}\left\langle x_{k} x_{l}\right\rangle &=\left\langle x_{l} A_{k}+x_{k} A_{l}+B_{k l}\right\rangle \end{aligned}$$

  [You may assume that $P(\mathbf{x}, t) \rightarrow 0$ as $|\mathbf{x}| \rightarrow \infty$.]

  (b) For small fluctuations, you may assume that the vector $\mathbf{A}$ may be approximated by a linear function in $\mathbf{x}$ and the matrix $\mathbf{B}$ may be treated as constant, i.e. $A_{k}(\mathbf{x}) \approx$ $a_{k l}\left(x_{l}-\left\langle x_{l}\right\rangle\right)$ and $B_{k l}(\mathbf{x}) \approx B_{k l}(\langle\mathbf{x}\rangle)=b_{k l}$ (where $a_{k l}$ and $b_{k l}$ are constants). Show that at steady state the covariances $C_{i j}=\operatorname{cov}\left(x_{i}, x_{j}\right)$ satisfy

  $$a_{i k} C_{j k}+a_{j k} C_{i k}+b_{i j}=0 .$$

  (c) A lab-controlled insect population consists of $x_{1}$ larvae and $x_{2}$ adults. Larvae are added to the system at rate $\lambda$. Larvae each mature at rate $\gamma$ per capita. Adults die at rate $\beta$ per capita. Give the vector $\mathbf{A}$ and matrix $\mathbf{B}$ for this model. Show that at steady state

  $$\left\langle x_{1}\right\rangle=\frac{\lambda}{\gamma}, \quad\left\langle x_{2}\right\rangle=\frac{\lambda}{\beta} .$$

  (d) Find the variance of each population size near steady state, and show that the covariance between the populations is zero.

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• Paper 3, Section II, E

  #Mathematical Biology

  2015

  A fungal disease is introduced into an isolated population of frogs. Without disease, the normalised population size $x$ would obey the logistic equation $\dot{x}=x(1-x)$, where the dot denotes differentiation with respect to time. The disease causes death at rate $d$ and there is no recovery. The disease transmission rate is $\beta$ and, in addition, offspring of infected frogs are infected from birth.

  (a) Briefly explain why the population sizes $x$ and $y$ of uninfected and infected frogs respectively now satisfy

  $$\begin{aligned} \dot{x} &=x[1-x-(1+\beta) y] \\ \dot{y} &=y[(1-d)-(1-\beta) x-y] \end{aligned}$$

  (b) The population starts at the disease-free population size $(x=1)$ and a small number of infected frogs are introduced. Show that the disease will successfully invade if and only if $\beta>d$.

  (c) By finding all the equilibria in $x \geqslant 0, y \geqslant 0$ and considering their stability, find the long-term outcome for the frog population. State the relationships between $d$ and $\beta$ that distinguish different final populations.

  (d) Plot the long-term steady total population size as a function of $d$ for fixed $\beta$, and note that an intermediate mortality rate is actually the most harmful. Explain why this is the case.

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• Paper 4, Section II, H

  #Number Fields

  2015

  Let $K$ be a number field. State Dirichlet's unit theorem, defining all the terms you use, and what it implies for a quadratic field $\mathbb{Q}(\sqrt{d})$, where $d \neq 0,1$ is a square-free integer.

  Find a fundamental unit of $\mathbb{Q}(\sqrt{26})$.

  Find all integral solutions $(x, y)$ of the equation $x^{2}-26 y^{2}=\pm 10$.

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• Paper 2, Section II, H

  #Number Fields

  2015

  (i) Let $d \equiv 2$ or $3 \bmod 4$. Show that $(p)$ remains prime in $\mathcal{O}_{\mathbb{Q}(\sqrt{d})}$ if and only if $x^{2}-d$ is irreducible $\bmod p$.

  (ii) Factorise $(2)$, (3) in $\mathcal{O}_{K}$, when $K=\mathbb{Q}(\sqrt{-14})$. Compute the class group of $K$.

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• Paper 1, Section II, $16 \mathrm{H}$

  #Number Fields

  2015

  (a) Let $K$ be a number field, and $f$ a monic polynomial whose coefficients are in $\mathcal{O}_{K}$. Let $M$ be a field containing $K$ and $\alpha \in M$. Show that if $f(\alpha)=0$, then $\alpha$ is an algebraic integer.

  Hence conclude that if $h \in K[x]$ is monic, with $h^{n} \in \mathcal{O}_{K}[x]$, then $h \in \mathcal{O}_{K}[x]$.

  (b) Compute an integral basis for $\mathcal{O}_{\mathbb{Q}(\alpha)}$ when the minimum polynomial of $\alpha$ is $x^{3}-x-4$.

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• Paper 4, Section I, H

  #Number Theory

  2015

  Show that if $10^{n}+1$ is prime then $n$ must be a power of 2 . Now assuming $n$ is a power of 2 , show that if $p$ is a prime factor of $10^{n}+1$ then $p \equiv 1(\bmod 2 n)$.

  Explain the method of Fermat factorization, and use it to factor $10^{4}+1$.

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• Paper 3, Section I, H

  #Number Theory

  2015

  What does it mean to say that a positive definite binary quadratic form is reduced? Find the three smallest positive integers properly represented by each of the forms $f(x, y)=3 x^{2}+8 x y+9 y^{2}$ and $g(x, y)=15 x^{2}+34 x y+20 y^{2}$. Show that every odd integer represented by some positive definite binary quadratic form with discriminant $-44$ is represented by at least one of the forms $f$ and $g$.

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• Paper 2, Section I, H

  #Number Theory

  2015

  Define the Euler totient function $\phi$ and the Möbius function $\mu$. Suppose $f$ and $g$ are functions defined on the natural numbers satisfying $f(n)=\sum_{d \mid n} g(d)$. State and prove a formula for $g$ in terms of $f$. Find a relationship between $\mu$ and $\phi$.

  Define the Riemann zeta function $\zeta(s)$. Find a Dirichlet series for $\zeta(s-1) / \zeta(s)$ valid for $\operatorname{Re}(s)>2$.

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• Paper 1, Section I, H

  #Number Theory

  2015

  Define the Legendre symbol $\left(\frac{a}{p}\right)$. State and prove Euler's criterion, assuming if you wish the existence of primitive roots $\bmod p$.

  By considering the prime factors of $n^{2}+4$ for $n$ an odd integer, prove that there are infinitely many primes $p$ with $p \equiv 5(\bmod 8)$.

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• Paper 4, Section II, H

  #Number Theory

  2015

  State the Chinese Remainder Theorem.

  Let $N$ be an odd positive integer. Define the Jacobi symbol $\left(\frac{a}{N}\right)$. Which of the following statements are true, and which are false? Give a proof or counterexample as appropriate.

  (i) If $\left(\frac{a}{N}\right)=1$ then the congruence $x^{2} \equiv a(\bmod N)$ is soluble.

  (ii) If $N$ is not a square then $\sum_{a=1}^{N}\left(\frac{a}{N}\right)=0$.

  (iii) If $N$ is composite then there exists an integer a coprime to $N$ with

  $$a^{N-1} \not \equiv 1 \quad(\bmod N)$$

  (iv) If $N$ is composite then there exists an integer $a$ coprime to $N$ with

  $$a^{(N-1) / 2} \not \equiv\left(\frac{a}{N}\right) \quad(\bmod N)$$

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• Paper 3, Section II, H

  #Number Theory

  2015

  Let $\theta$ be a real number with continued fraction expansion $\left[a_{0}, a_{1}, a_{2}, \ldots\right]$. Define the convergents $p_{n} / q_{n}$ (by means of recurrence relations) and show that for $\beta>0$ we have

  $$\left[a_{0}, a_{1}, \ldots, a_{n-1}, \beta\right]=\frac{\beta p_{n-1}+p_{n-2}}{\beta q_{n-1}+q_{n-2}}$$

  Show that

  $$\left|\theta-\frac{p_{n}}{q_{n}}\right|<\frac{1}{q_{n} q_{n+1}}$$

  and deduce that $p_{n} / q_{n} \rightarrow \theta$ as $n \rightarrow \infty$.

  By computing a suitable continued fraction expansion, find solutions in positive integers $x$ and $y$ to each of the equations $x^{2}-53 y^{2}=4$ and $x^{2}-53 y^{2}=-7$.

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• Paper 4, Section II, E

  #Numerical Analysis

  2015

  (a) Define the $m$ th Krylov space $K_{m}(A, v)$ for $A \in \mathbb{R}^{n \times n}$ and $0 \neq v \in \mathbb{R}^{n}$. Letting $\delta_{m}$ be the dimension of $K_{m}(A, v)$, prove the following results.

  (i) There exists a positive integer $s \leqslant n$ such that $\delta_{m}=m$ for $m \leqslant s$ and $\delta_{m}=s$ for $m>s$.

  (ii) If $v=\sum_{i=1}^{s^{\prime}} c_{i} w_{i}$, where $w_{i}$ are eigenvectors of $A$ for distinct eigenvalues and all $c_{i}$ are nonzero, then $s=s^{\prime}$.

  (b) Define the term residual in the conjugate gradient (CG) method for solving a system $A x=b$ with symmetric positive definite $A$. Explain (without proof) the connection to Krylov spaces and prove that for any right-hand side $b$ the CG method finds an exact solution after at most $t$ steps, where $t$ is the number of distinct eigenvalues of $A$. [You may use without proof known properties of the iterates of the CG method.]

  Define what is meant by preconditioning, and explain two ways in which preconditioning can speed up convergence. Can we choose the preconditioner so that the CG method requires only one step? If yes, is it a reasonable method for speeding up the computation?

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• Paper 2, Section II, E

  #Numerical Analysis

  2015

  (a) The boundary value problem $-\Delta u+c u=f$ on the unit square $[0,1]^{2}$ with zero boundary conditions and scalar constant $c>0$ is discretised using finite differences as

  $$\begin{array}{r} -u_{i-1, j}-u_{i+1, j}-u_{i, j-1}-u_{i, j+1}+4 u_{i, j}+c h^{2} u_{i, j}=h^{2} f(i h, j h), \\ i, j=1, \ldots, m \end{array}$$

  with $h=1 /(m+1)$. Show that for the resulting system $A u=b$, for a suitable matrix $A$ and vectors $u$ and $b$, both the Jacobi and Gauss-Seidel methods converge. [You may cite and use known results on the discretised Laplace operator and on the convergence of iterative methods.]

  Define the Jacobi method with relaxation parameter $\omega$. Find the eigenvalues $\lambda_{k, l}$ of the iteration matrix $H_{\omega}$ for the above problem and show that, in order to ensure convergence for all $h$, the condition $0<\omega \leqslant 1$ is necessary.

  [Hint: The eigenvalues of the discretised Laplace operator in two dimensions are $4\left(\sin ^{2} \frac{\pi k h}{2}+\sin ^{2} \frac{\pi l h}{2}\right)$ for integers $k, l$.]

  (b) Explain the components and steps in a multigrid method for solving the Poisson equation, discretised as $A_{h} u_{h}=b_{h}$. If we use the relaxed Jacobi method within the multigrid method, is it necessary to choose $\omega \neq 1$ to get fast convergence? Explain why or why not.

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• Paper 3, Section II, E

  #Numerical Analysis

  2015

  (a) Given the finite-difference recurrence

  $$\sum_{k=r}^{s} a_{k} u_{m+k}^{n+1}=\sum_{k=r}^{s} b_{k} u_{m+k}^{n}, \quad m \in \mathbb{Z}, n \in \mathbb{Z}^{+}$$

  that discretises a Cauchy problem, the amplification factor is defined by

  $$H(\theta)=\left(\sum_{k=r}^{s} b_{k} e^{i k \theta}\right) /\left(\sum_{k=r}^{s} a_{k} e^{i k \theta}\right)$$

  Show how $H(\theta)$ acts on the Fourier transform $\hat{u}^{n}$ of $u^{n}$. Hence prove that the method is stable if and only if $|H(\theta)| \leqslant 1$ for all $\theta \in[-\pi, \pi]$.

  (b) The two-dimensional diffusion equation

  $$u_{t}=u_{x x}+c u_{y y}$$

  for some scalar constant $c>0$ is discretised with the forward Euler scheme

  $$u_{i, j}^{n+1}=u_{i, j}^{n}+\mu\left(u_{i+1, j}^{n}-2 u_{i, j}^{n}+u_{i-1, j}^{n}+c u_{i, j+1}^{n}-2 c u_{i, j}^{n}+c u_{i, j-1}^{n}\right)$$

  Using Fourier stability analysis, find the range of values $\mu>0$ for which the scheme is stable.

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• Paper 4, Section II, $25 K$

  #Optimization and Control

  2015

  Consider the scalar system evolving as

  $$x_{t}=x_{t-1}+u_{t-1}+\epsilon_{t}, \quad t=1,2, \ldots,$$

  where $\left\{\epsilon_{t}\right\}_{t=1}^{\infty}$ is a white noise sequence with $E \epsilon_{t}=0$ and $E \epsilon_{t}^{2}=v$. It is desired to choose controls $\left\{u_{t}\right\}_{t=0}^{h-1}$ to minimize $E\left[\sum_{t=0}^{h-1}\left(\frac{1}{2} x_{t}^{2}+u_{t}^{2}\right)+x_{h}^{2}\right]$. Show that for $h=6$ the minimal cost is $x_{0}^{2}+6 v$.

  Find a constant $\lambda$ and a function $\phi$ which solve

  $$\phi(x)+\lambda=\min _{u}\left[\frac{1}{2} x^{2}+u^{2}+E \phi\left(x+u+\epsilon_{1}\right)\right]$$

  Let $P$ be the class of those policies for which every $u_{t}$ obeys the constraint $\left(x_{t}+u_{t}\right)^{2} \leqslant(0.9) x_{t}^{2}$. Show that $E_{\pi} \phi\left(x_{t}\right) \leqslant x_{0}^{2}+10 v$, for all $\pi \in P$. Find, and prove optimal, a policy which over all $\pi \in P$ minimizes

  $$\lim _{h \rightarrow \infty} \frac{1}{h} E_{\pi}\left[\sum_{t=0}^{h-1}\left(\frac{1}{2} x_{t}^{2}+u_{t}^{2}\right)\right]$$

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• Paper 3, Section II, K

  #Optimization and Control

  2015

  A burglar having wealth $x$ may retire, or go burgling another night, in either of towns 1 or 2 . If he burgles in town $i$ then with probability $p_{i}=1-q_{i}$ he will, independently of previous nights, be caught, imprisoned and lose all his wealth. If he is not caught then his wealth increases by 0 or $2 a_{i}$, each with probability $1 / 2$ and independently of what happens on other nights. Values of $p_{i}$ and $a_{i}$ are the same every night. He wishes to maximize his expected wealth at the point he retires, is imprisoned, or $s$ nights have elapsed.

  Using the dynamic programming equation

  $$F_{s}(x)=\max \left\{x, q_{1} E F_{s-1}\left(x+R_{1}\right), q_{2} E F_{s-1}\left(x+R_{2}\right)\right\}$$

  with $R_{j}, F_{0}(x)$ appropriately defined, prove that there exists an optimal policy under which he burgles another night if and only if his wealth is less than $x^{*}=\max _{i}\left\{a_{i} q_{i} / p_{i}\right\}$.

  Suppose $q_{1}>q_{2}$ and $q_{1} a_{1}>q_{2} a_{2}$. Prove that he should never burgle in town 2 .

  [Hint: Suppose $x<x^{*}$, there are $s$ nights to go, and it has been shown that he ought not burgle in town 2 if less than $s$ nights remain. For the case $a_{2}>a_{1}$, separately consider subcases $x+2 a_{2} \geqslant x^{*}$ and $x+2 a_{2}<x^{*}$. An interchange argument may help.]

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• Paper 1, Section II, 38E

  #Numerical Analysis

  2015

  (a) The diffusion equation

  $$\frac{\partial u}{\partial t}=\frac{\partial}{\partial x}\left(a(x) \frac{\partial u}{\partial x}\right) \quad \text { in } \quad 0 \leqslant x \leqslant 1, \quad t \geqslant 0$$

  with the initial condition $u(x, 0)=\phi(x)$ in $0 \leqslant x \leqslant 1$ and zero boundary conditions at $x=0$ and $x=1$, is solved by the finite-difference method

  $$\begin{array}{r} u_{m}^{n+1}=u_{m}^{n}+\mu\left[a_{m-\frac{1}{2}} u_{m-1}^{n}-\left(a_{m-\frac{1}{2}}+a_{m+\frac{1}{2}}\right) u_{m}^{n}+a_{m+\frac{1}{2}} u_{m+1}^{n}\right] \\ m=1,2, \ldots, M \end{array}$$

  where $\mu=k / h^{2}, k=\Delta t, h=1 /(M+1), u_{m}^{n} \approx u(m h, n k)$, and $a_{\alpha}=a(\alpha h)$.

  Assuming that the function $a$ and the exact solution are sufficiently smooth, prove that the exact solution satisfies the numerical scheme with error $O\left(h^{3}\right)$ for constant $\mu$.

  (b) For the problem in part (a), assume that there exist $0<a_{-}<a_{+}<\infty$ such that $a_{-} \leqslant a(x) \leqslant a_{+}$for all $0 \leqslant x \leqslant 1$. State (without proof) the Gershgorin theorem and prove that the method is stable for $0<\mu \leqslant 1 /\left(2 a_{+}\right)$.

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• Paper 2, Section II, $26 \mathrm{~K}$

  #Optimization and Control

  2015

  As a function of policy $\pi$ and initial state $x$, let

  $$F(\pi, x)=E_{\pi}\left[\sum_{t=0}^{\infty} \beta^{t} r\left(x_{t}, u_{t}\right) \mid x_{0}=x\right]$$

  where $\beta \geqslant 1$ and $r(x, u) \geqslant 0$ for all $x, u$. Suppose that for a specific policy $\pi$, and all $x$,

  $$F(\pi, x)=\sup _{u}\left\{r(x, u)+\beta E\left[F\left(\pi, x_{1}\right) \mid x_{0}=x, u_{0}=u\right]\right\} .$$

  Prove that $F(\pi, x) \geqslant F\left(\pi^{\prime}, x\right)$ for all $\pi^{\prime}$ and $x$.

  A gambler plays games in which he may bet 1 or 2 pounds, but no more than his present wealth. Suppose he has $x_{t}$ pounds after $t$ games. If he bets $i$ pounds then $x_{t+1}=x_{t}+i$, or $x_{t+1}=x_{t}-i$, with probabilities $p_{i}$ and $1-p_{i}$ respectively. Gambling terminates at the first $\tau$ such that $x_{\tau}=0$ or $x_{\tau}=100$. His final reward is $(9 / 8)^{\tau / 2} x_{\tau}$. Let $\pi$ be the policy of always betting 1 pound. Given $p_{1}=1 / 3$, show that $F(\pi, x) \propto x 2^{x / 2}$.

  Is $\pi$ optimal when $p_{2}=1 / 4$ ?

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• Paper 4, Section II, E

  #Partial Differential Equations

  2015

  (a) Show that the Cauchy problem for $u(x, t)$ satisfying

  $$u_{t}+u=u_{x x}$$

  with initial data $u(x, 0)=u_{0}(x)$, which is a smooth $2 \pi$-periodic function of $x$, defines a strongly continuous one parameter semi-group of contractions on the Sobolev space $H_{\text {per }}^{s}$ for any $s \in\{0,1,2, \ldots\}$.

  (b) Solve the Cauchy problem for the equation

  $$u_{t t}+u_{t}+\frac{1}{4} u=u_{x x}$$

  with $u(x, 0)=u_{0}(x), u_{t}(x, 0)=u_{1}(x)$, where $u_{0}, u_{1}$ are smooth $2 \pi$-periodic functions of $x$, and show that the solution is smooth. Prove from first principles that the solution satisfies the property of finite propagation speed.

  [In this question all functions are real-valued, and

  $$H_{\text {per }}^{s}=\left\{u=\sum_{m \in \mathbb{Z}} \hat{u}(m) e^{i m x} \in L^{2}:\|u\|_{H^{s}}^{2}=\sum_{m \in \mathbb{Z}}\left(1+m^{2}\right)^{s}|\hat{u}(m)|^{2}<\infty\right\}$$

  are the Sobolev spaces of functions which are $2 \pi$-periodic in $x$, for $s=0,1,2, \ldots]$

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• Paper 3, Section II, E

  #Partial Differential Equations

  2015

  (a) Show that if $f \in \mathcal{S}\left(\mathbb{R}^{n}\right)$ is a Schwartz function and $u$ is a tempered distribution which solves

  $$-\Delta u+m^{2} u=f$$

  for some constant $m \neq 0$, then there exists a number $C>0$ which depends only on $m$, such that $\|u\|_{H^{s+2}} \leqslant C\|f\|_{H^{s}}$ for any $s \geqslant 0$. Explain briefly why this inequality remains valid if $f$ is only assumed to be in $H^{s}\left(\mathbb{R}^{n}\right)$.

  Show that if $\epsilon>0$ is given then $\|v\|_{H^{1}}^{2} \leqslant \epsilon\|v\|_{H^{2}}^{2}+\frac{1}{4 \epsilon}\|v\|_{H^{0}}^{2}$ for any $v \in H^{2}\left(\mathbb{R}^{n}\right)$.

  [Hint: The inequality $a \leqslant \epsilon a^{2}+\frac{1}{4 \epsilon}$ holds for any positive $\epsilon$ and $a \in \mathbb{R} .$ ]

  Prove that if $u$ is a smooth bounded function which solves

  $$-\Delta u+m^{2} u=u^{3}+\alpha \cdot \nabla u$$

  for some constant vector $\alpha \in \mathbb{R}^{n}$ and constant $m \neq 0$, then there exists a number $C^{\prime}>0$ such that $\|u\|_{H^{2}} \leqslant C^{\prime}$ and $C^{\prime}$ depends only on $m, \alpha,\|u\|_{L^{\infty}},\|u\|_{L^{2}}$.

  [You may use the fact that, for non-negative $s$, the Sobolev space of functions

  $$\left.H^{s}\left(\mathbb{R}^{n}\right)=\left\{f \in L^{2}\left(\mathbb{R}^{n}\right):\|f\|_{H^{s}}^{2}=\int_{\mathbb{R}^{n}}\left(1+\|\xi\|^{2}\right)^{s}|\hat{f}(\xi)|^{2} d \xi<\infty\right\} .\right]$$

  (b) Let $u(x, t)$ be a smooth real-valued function, which is $2 \pi$-periodic in $x$ and satisfies the equation

  $$u_{t}=u^{2} u_{x x}+u^{3}$$

  Give a complete proof that if $u(x, 0)>0$ for all $x$ then $u(x, t)>0$ for all $x$ and $t>0$.

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• Paper 2, Section II, E

  #Partial Differential Equations

  2015

  Prove that if $\phi \in C\left(\mathbb{R}^{n}\right)$ is absolutely integrable with $\int \phi(x) d x=1$, and $\phi_{\epsilon}(x)=\epsilon^{-n} \phi(x / \epsilon)$ for $\epsilon>0$, then for every Schwartz function $f \in \mathcal{S}\left(\mathbb{R}^{n}\right)$ the convolution

  $$\phi_{\epsilon} * f(x) \rightarrow f(x)$$

  uniformly in $x$ as $\epsilon \downarrow 0$.

  Show that the function $N_{\epsilon} \in C^{\infty}\left(\mathbb{R}^{3}\right)$ given by

  $$N_{\epsilon}(x)=\frac{1}{4 \pi \sqrt{|x|^{2}+\epsilon^{2}}}$$

  for $\epsilon>0$ satisfies

  $$\lim _{\epsilon \rightarrow 0} \int_{\mathbb{R}^{3}}-\Delta N_{\epsilon}(x) f(x) d x=f(0)$$

  for $f \in \mathcal{S}\left(\mathbb{R}^{n}\right)$. Hence prove that the tempered distribution determined by the function $N(x)=(4 \pi|x|)^{-1}$ is a fundamental solution of the operator $-\Delta .$

  [You may use the fact that $\int_{0}^{\infty} r^{2} /\left(1+r^{2}\right)^{5 / 2} d r=1 / 3 .$ ]

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• Paper 1, Section II, E

  #Partial Differential Equations

  2015

  (a) State the Cauchy-Kovalevskaya theorem, and explain for which values of $a \in \mathbb{R}$ it implies the existence of solutions to the Cauchy problem

  $$x u_{x}+y u_{y}+a u_{z}=u, \quad u(x, y, 0)=f(x, y),$$

  where $f$ is real analytic. Using the method of characteristics, solve this problem for these values of $a$, and comment on the behaviour of the characteristics as $a$ approaches any value where the non-characteristic condition fails.

  (b) Consider the Cauchy problem

  $$u_{y}=v_{x}, \quad v_{y}=-u_{x}$$

  with initial data $u(x, 0)=f(x)$ and $v(x, 0)=0$ which are $2 \pi$-periodic in $x$. Give an example of a sequence of smooth solutions $\left(u_{n}, v_{n}\right)$ which are also $2 \pi$-periodic in $x$ whose corresponding initial data $u_{n}(x, 0)=f_{n}(x)$ and $v_{n}(x, 0)=0$ are such that $\int_{0}^{2 \pi}\left|f_{n}(x)\right|^{2} d x \rightarrow 0$ while $\int_{0}^{2 \pi}\left|u_{n}(x, y)\right|^{2} d x \rightarrow \infty$ for non-zero $y$ as $n \rightarrow \infty$

  Comment on the significance of this in relation to the concept of well-posedness.

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• Paper 4, Section II, A

  #Principles of Quantum Mechanics

  2015

  The Hamiltonian for a quantum system in the Schrödinger picture is $H_{0}+\lambda V(t)$, where $H_{0}$ is independent of time and the parameter $\lambda$ is small. Define the interaction picture corresponding to this Hamiltonian and derive a time evolution equation for interaction picture states.

  Suppose that $|\chi\rangle$ and $|\phi\rangle$ are eigenstates of $H_{0}$ with distinct eigenvalues $E$ and $E^{\prime}$, respectively. Show that if the system is in state $|\chi\rangle$ at time zero then the probability of measuring it to be in state $|\phi\rangle$ at time $t$ is

  $$\frac{\lambda^{2}}{\hbar^{2}}\left|\int_{0}^{t} d t^{\prime}\left\langle\phi\left|V\left(t^{\prime}\right)\right| \chi\right\rangle e^{i\left(E^{\prime}-E\right) t^{\prime} / \hbar}\right|^{2}+O\left(\lambda^{3}\right)$$

  Let $H_{0}$ be the Hamiltonian for an isotropic three-dimensional harmonic oscillator of mass $m$ and frequency $\omega$, with $\chi(r)$ being the ground state wavefunction (where $r=|\mathbf{x}|$ ) and $\phi_{i}(\mathbf{x})=(2 m \omega / \hbar)^{1 / 2} x_{i} \chi(r)$ being wavefunctions for the states at the first excited energy level $(i=1,2,3)$. The oscillator is in its ground state at $t=0$ when a perturbation

  $$\lambda V(t)=\lambda \hat{x}_{3} e^{-\mu t}$$

  is applied, with $\mu>0$, and $H_{0}$ is then measured after a very large time has elapsed. Show that to first order in perturbation theory the oscillator will be found in one particular state at the first excited energy level with probability

  $$\frac{\lambda^{2}}{2 \hbar m \omega\left(\mu^{2}+\omega^{2}\right)},$$

  but that the probability that it will be found in either of the other excited states is zero (to this order).

  $\left[\right.$ You may use the fact that $\left.4 \pi \int_{0}^{\infty} r^{4}|\chi(r)|^{2} d r=\frac{3 \hbar}{2 m \omega} .\right]$

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• Paper 3, Section II, A

  #Principles of Quantum Mechanics

  2015

  Let $|j, m\rangle$ denote the normalised joint eigenstates of $\mathbf{J}^{2}$ and $J_{3}$, where $\mathbf{J}$ is the angular momentum operator for a quantum system. State clearly the possible values of the quantum numbers $j$ and $m$ and write down the corresponding eigenvalues in units with $\hbar=1$.

  Consider two quantum systems with angular momentum states $\left|\frac{1}{2}, r\right\rangle$ and $|j, m\rangle$. The eigenstates corresponding to their combined angular momentum can be written as

  $$|J, M\rangle=\sum_{r, m} C_{r m}^{J M}\left|\frac{1}{2}, r\right\rangle|j, m\rangle,$$

  where $C_{r m}^{J M}$ are Clebsch-Gordan coefficients for addition of angular momenta $\frac{1}{2}$ and $j$. What are the possible values of $J$ and what is a necessary condition relating $r, m$ and $M$ in order that $C_{r m}^{J M} \neq 0$ ?

  Calculate the values of $C_{r m}^{J M}$ for $j=2$ and for all $M \geqslant \frac{3}{2}$. Use the sign convention that $C_{r m}^{J J}>0$ when $m$ takes its maximum value.

  A particle $X$ with spin $\frac{3}{2}$ and intrinsic parity $\eta_{X}$ is at rest. It decays into two particles $A$ and $B$ with spin $\frac{1}{2}$ and spin 0 , respectively. Both $A$ and $B$ have intrinsic parity $-1$. The relative orbital angular momentum quantum number for the two particle system is $\ell$. What are the possible values of $\ell$ for the cases $\eta_{X}=+1$ and $\eta_{X}=-1$ ?

  Suppose particle $X$ is prepared in the state $\left|\frac{3}{2}, \frac{3}{2}\right\rangle$ before it decays. Calculate the probability $P$ for particle $A$ to be found in the state $\left|\frac{1}{2}, \frac{1}{2}\right\rangle$, given that $\eta_{X}=+1$.

  What is the probability $P$ if instead $\eta_{X}=-1$ ?

  [Units with $\hbar=1$ should be used throughout. You may also use without proof

  $$\left.J_{-}|j, m\rangle=\sqrt{(j+m)(j-m+1)}|j, m-1\rangle .\right]$$

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• Paper 2, Section II, A

  #Principles of Quantum Mechanics

  2015

  Express the spin operator $\mathbf{S}$ for a particle of spin $\frac{1}{2}$ in terms of the Pauli matrices $\boldsymbol{\sigma}=\left(\sigma_{1}, \sigma_{2}, \sigma_{3}\right)$ where

  $$\sigma_{1}=\left(\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right), \quad \sigma_{2}=\left(\begin{array}{cc} 0 & -i \\ i & 0 \end{array}\right), \quad \sigma_{3}=\left(\begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array}\right)$$

  Show that $(\mathbf{n} \cdot \boldsymbol{\sigma})^{2}=\mathbb{I}$ for any unit vector $\mathbf{n}$ and deduce that

  $$e^{-i \theta \mathbf{n} \cdot \mathbf{S} / \hbar}=\mathbb{I} \cos (\theta / 2)-i(\mathbf{n} \cdot \boldsymbol{\sigma}) \sin (\theta / 2) .$$

  The space of states $V$ for a particle of spin $\frac{1}{2}$ has basis states $|\uparrow\rangle,|\downarrow\rangle$ which are eigenstates of $S_{3}$ with eigenvalues $\frac{1}{2} \hbar$ and $-\frac{1}{2} \hbar$ respectively. If the Hamiltonian for the particle is $H=\frac{1}{2} \alpha \hbar \sigma_{1}$, find

  $$e^{-i t H / \hbar}|\uparrow\rangle \quad \text { and } \quad e^{-i t H / \hbar}|\downarrow\rangle$$

  as linear combinations of the basis states.

  The space of states for a system of two spin $\frac{1}{2}$ particles is $V \otimes V$. Write down explicit expressions for the joint eigenstates of $\mathbf{J}^{2}$ and $J_{3}$, where $\mathbf{J}$ is the sum of the spin operators for the particles.

  Suppose that the two-particle system has Hamiltonian $H=\frac{1}{2} \lambda \hbar\left(\sigma_{1} \otimes \mathbb{I}-\mathbb{I} \otimes \sigma_{1}\right)$ and that at time $t=0$ the system is in the state with $J_{3}$ eigenvalue $\hbar$. Calculate the probability that at time $t>0$ the system will be measured to be in the state with $\mathbf{J}^{2}$ eigenvalue zero.

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• Paper 1, Section II, A

  #Principles of Quantum Mechanics

  2015

  If $A$ and $B$ are operators which each commute with their commutator $[A, B]$, show that

  $$F(\lambda)=e^{\lambda A} e^{\lambda B} e^{-\lambda(A+B)} \quad \text { satisfies } \quad F^{\prime}(\lambda)=\lambda[A, B] F(\lambda)$$

  By solving this differential equation for $F(\lambda)$, deduce that

  $$e^{A} e^{B}=e^{\frac{1}{2}[A, B]} e^{A+B}$$

  The annihilation and creation operators for a harmonic oscillator of mass $m$ and frequency $\omega$ are defined by

  $$a=\sqrt{\frac{m \omega}{2 \hbar}}\left(\hat{x}+\frac{i}{m \omega} \hat{p}\right), \quad a^{\dagger}=\sqrt{\frac{m \omega}{2 \hbar}}\left(\hat{x}-\frac{i}{m \omega} \hat{p}\right)$$

  Write down an expression for the general normalised eigenstate $|n\rangle(n=0,1,2, \ldots)$ of the oscillator Hamiltonian $H$ in terms of the ground state $|0\rangle$. What is the energy eigenvalue $E_{n}$ of the state $|n\rangle ?$

  Suppose the oscillator is now subject to a small perturbation so that it is described by the modified Hamiltonian $H+\varepsilon V(\hat{x})$ with $V(\hat{x})=\cos (\mu \hat{x})$. Show that

  $$V(\hat{x})=\frac{1}{2} e^{-\gamma^{2} / 2}\left(e^{i \gamma a^{\dagger}} e^{i \gamma a}+e^{-i \gamma a^{\dagger}} e^{-i \gamma a}\right)$$

  where $\gamma$ is a constant, to be determined. Hence show that to $O\left(\varepsilon^{2}\right)$ the shift in the ground state energy as a result of the perturbation is

  $$\varepsilon e^{-\mu^{2} \hbar / 4 m \omega}-\varepsilon^{2} e^{-\mu^{2} \hbar / 2 m \omega} \frac{1}{\hbar \omega} \sum_{p=1}^{\infty} \frac{1}{(2 p) ! 2 p}\left(\frac{\mu^{2} \hbar}{2 m \omega}\right)^{2 p} .$$

  [Standard results of perturbation theory may be quoted without proof.]

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• Paper 4, Section II, $\mathbf{2 4 J}$

  #Principles of Statistics

  2015

  Given independent and identically distributed observations $X_{1}, \ldots, X_{n}$ with finite mean $E\left(X_{1}\right)=\mu$ and variance $\operatorname{Var}\left(X_{1}\right)=\sigma^{2}$, explain the notion of a bootstrap sample $X_{1}^{b}, \ldots, X_{n}^{b}$, and discuss how you can use it to construct a confidence interval $C_{n}$ for $\mu$.

  Suppose you can operate a random number generator that can simulate independent uniform random variables $U_{1}, \ldots, U_{n}$ on $[0,1]$. How can you use such a random number generator to simulate a bootstrap sample?

  Suppose that $\left(F_{n}: n \in \mathbb{N}\right)$ and $F$ are cumulative probability distribution functions defined on the real line, that $F_{n}(t) \rightarrow F(t)$ as $n \rightarrow \infty$ for every $t \in \mathbb{R}$, and that $F$ is continuous on $\mathbb{R}$. Show that, as $n \rightarrow \infty$,

  $$\sup _{t \in \mathbb{R}}\left|F_{n}(t)-F(t)\right| \rightarrow 0 .$$

  State (without proof) the theorem about the consistency of the bootstrap of the mean, and use it to give an asymptotic justification of the confidence interval $C_{n}$. That is, prove that as $n \rightarrow \infty, P^{\mathbb{N}}\left(\mu \in C_{n}\right) \rightarrow 1-\alpha$ where $P^{\mathbb{N}}$ is the joint distribution of $X_{1}, X_{2}, \ldots$

  [You may use standard facts of stochastic convergence and the Central Limit Theorem without proof.]

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• Paper 3, Section II, J

  #Principles of Statistics

  2015

  Define what it means for an estimator $\hat{\theta}$ of an unknown parameter $\theta$ to be consistent.

  Let $S_{n}$ be a sequence of random real-valued continuous functions defined on $\mathbb{R}$ such that, as $n \rightarrow \infty, S_{n}(\theta)$ converges to $S(\theta)$ in probability for every $\theta \in \mathbb{R}$, where $S: \mathbb{R} \rightarrow \mathbb{R}$ is non-random. Suppose that for some $\theta_{0} \in \mathbb{R}$ and every $\varepsilon>0$ we have

  $$S\left(\theta_{0}-\varepsilon\right)<0<S\left(\theta_{0}+\varepsilon\right),$$

  and that $S_{n}$ has exactly one zero $\hat{\theta}_{n}$ for every $n \in \mathbb{N}$. Show that $\hat{\theta}_{n} \rightarrow \theta_{0}$ as $n \rightarrow \infty$, and deduce from this that the maximum likelihood estimator (MLE) based on observations $X_{1}, \ldots, X_{n}$ from a $N(\theta, 1), \theta \in \mathbb{R}$ model is consistent.

  Now consider independent observations $\mathbf{X}_{1}, \ldots, \mathbf{X}_{n}$ of bivariate normal random vectors

  $$\mathbf{X}_{i}=\left(X_{1 i}, X_{2 i}\right)^{T} \sim N_{2}\left[\left(\mu_{i}, \mu_{i}\right)^{T}, \sigma^{2} I_{2}\right], \quad i=1, \ldots, n,$$

  where $\mu_{i} \in \mathbb{R}, \sigma>0$ and $I_{2}$ is the $2 \times 2$ identity matrix. Find the MLE $\hat{\mu}=\left(\hat{\mu}_{1}, \ldots, \hat{\mu}_{n}\right)^{T}$ of $\mu=\left(\mu_{1}, \ldots, \mu_{n}\right)^{T}$ and show that the MLE of $\sigma^{2}$ equals

  $$\hat{\sigma}^{2}=\frac{1}{n} \sum_{i=1}^{n} s_{i}^{2}, \quad s_{i}^{2}=\frac{1}{2}\left[\left(X_{1 i}-\hat{\mu}_{i}\right)^{2}+\left(X_{2 i}-\hat{\mu}_{i}\right)^{2}\right]$$

  Show that $\hat{\sigma}^{2}$ is not consistent for estimating $\sigma^{2}$. Explain briefly why the MLE fails in this model.

  [You may use the Law of Large Numbers without proof.]

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• Paper 2, Section II, J

  #Principles of Statistics

  2015

  Consider a random variable $X$ arising from the binomial distribution $\operatorname{Bin}(n, \theta)$, $\theta \in \Theta=[0,1]$. Find the maximum likelihood estimator $\hat{\theta}_{M L E}$ and the Fisher information $I(\theta)$ for $\theta \in \Theta$.

  Now consider the following priors on $\Theta$ :

  (i) a uniform $U([0,1])$ prior on $[0,1]$,

  (ii) a prior with density $\pi(\theta)$ proportional to $\sqrt{I(\theta)}$,

  (iii) a $\operatorname{Beta}(\sqrt{n} / 2, \sqrt{n} / 2)$ prior.

  Find the means $E[\theta \mid X]$ and modes $m_{\theta} \mid X$ of the posterior distributions corresponding to the prior distributions (i)-(iii). Which of these posterior decision rules coincide with $\hat{\theta}_{M L E}$ ? Which one is minimax for quadratic risk? Justify your answers.

  [You may use the following properties of the $\operatorname{Beta}(a, b)(a>0, b>0)$ distribution. Its density $f(x ; a, b), x \in[0,1]$, is proportional to $x^{a-1}(1-x)^{b-1}$, its mean is equal to $a /(a+b)$, and its mode is equal to

  $$\frac{\max (a-1,0)}{\max (a, 1)+\max (b, 1)-2}$$

  provided either $a>1$ or $b>1$.

  You may further use the fact that a unique Bayes rule of constant risk is a unique minimax rule for that risk.]

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• Paper 1, Section II, J

  #Principles of Statistics

  2015

  Consider a normally distributed random vector $X \in \mathbb{R}^{p}$ modelled as $X \sim N\left(\theta, I_{p}\right)$ where $\theta \in \mathbb{R}^{p}, I_{p}$ is the $p \times p$ identity matrix, and where $p \geqslant 3$. Define the Stein estimator $\hat{\theta}_{S T E I N}$ of $\theta$.

  Prove that $\hat{\theta}_{S T E I N}$ dominates the estimator $\tilde{\theta}=X$ for the risk function induced by quadratic loss

  $$\ell(a, \theta)=\sum_{i=1}^{p}\left(a_{i}-\theta_{i}\right)^{2}, \quad a \in \mathbb{R}^{p}$$

  Show however that the worst case risks coincide, that is, show that

  $$\sup _{\theta \in \mathbb{R}^{p}} E_{\theta} \ell(X, \theta)=\sup _{\theta \in \mathbb{R}^{p}} E_{\theta} \ell\left(\hat{\theta}_{S T E I N}, \theta\right)$$

  [You may use Stein's lemma without proof, provided it is clearly stated.]

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• Paper 4, Section II, J

  #Probability and Measure

  2015

  (a) State Fatou's lemma.

  (b) Let $X$ be a random variable on $\mathbb{R}^{d}$ and let $\left(X_{k}\right)_{k=1}^{\infty}$ be a sequence of random variables on $\mathbb{R}^{d}$. What does it mean to say that $X_{k} \rightarrow X$ weakly?

  State and prove the Central Limit Theorem for i.i.d. real-valued random variables. [You may use auxiliary theorems proved in the course provided these are clearly stated.]

  (c) Let $X$ be a real-valued random variable with characteristic function $\varphi$. Let $\left(h_{n}\right)_{n=1}^{\infty}$ be a sequence of real numbers with $h_{n} \neq 0$ and $h_{n} \rightarrow 0$. Prove that if we have

  $$\liminf _{n \rightarrow \infty} \frac{2 \varphi(0)-\varphi\left(-h_{n}\right)-\varphi\left(h_{n}\right)}{h_{n}^{2}}<\infty$$

  then $\mathbb{E}\left[X^{2}\right]<\infty$

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• Paper 3 , Section II, J

  #Probability and Measure

  2015

  (a) Let $(E, \mathcal{E}, \mu)$ be a measure space. What does it mean to say that $T: E \rightarrow E$ is a measure-preserving transformation? What does it mean to say that a set $A \in \mathcal{E}$ is invariant under $T$ ? Show that the class of invariant sets forms a $\sigma$-algebra.

  (b) Take $E$ to be $[0,1)$ with Lebesgue measure on its Borel $\sigma$-algebra. Show that the baker's map $T:[0,1) \rightarrow[0,1)$ defined by

  $$T(x)=2 x-\lfloor 2 x\rfloor$$

  is measure-preserving.

  (c) Describe in detail the construction of the canonical model for sequences of independent random variables having a given distribution $m$.

  Define the Bernoulli shift map and prove it is a measure-preserving ergodic transformation.

  [You may use without proof other results concerning sequences of independent random variables proved in the course, provided you state these clearly.]

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• Paper 2, Section II, J

  #Probability and Measure

  2015

  (a) Let $(E, \mathcal{E}, \mu)$ be a measure space, and let $1 \leqslant p<\infty$. What does it mean to say that $f$ belongs to $L^{p}(E, \mathcal{E}, \mu)$ ?

  (b) State Hölder's inequality.

  (c) Consider the measure space of the unit interval endowed with Lebesgue measure. Suppose $f \in L^{2}(0,1)$ and let $0<\alpha<1 / 2$.

  (i) Show that for all $x \in \mathbb{R}$,

  $$\int_{0}^{1}|f(y)||x-y|^{-\alpha} d y<\infty$$

  (ii) For $x \in \mathbb{R}$, define

  $$g(x)=\int_{0}^{1} f(y)|x-y|^{-\alpha} d y$$

  Show that for $x \in \mathbb{R}$ fixed, the function $g$ satisfies

  $$|g(x+h)-g(x)| \leqslant\|f\|_{2} \cdot(I(h))^{1 / 2},$$

  where

  $$I(h)=\int_{0}^{1}\left(|x+h-y|^{-\alpha}-|x-y|^{-\alpha}\right)^{2} d y .$$

  (iii) Prove that $g$ is a continuous function. [Hint: You may find it helpful to split the integral defining $I(h)$ into several parts.]

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• Paper 1, Section II, J

  #Probability and Measure

  2015

  (a) Define the following concepts: a $\pi$-system, a $d$-system and a $\sigma$-algebra.

  (b) State the Dominated Convergence Theorem.

  (c) Does the set function

  $$\mu(A)= \begin{cases}0 & \text { for } A \text { bounded } \\ 1 & \text { for } A \text { unbounded }\end{cases}$$

  furnish an example of a Borel measure?

  (d) Suppose $g:[0,1] \rightarrow[0,1]$ is a measurable function. Let $f:[0,1] \rightarrow \mathbb{R}$ be continuous with $f(0) \leqslant f(1)$. Show that the limit

  $$\lim _{n \rightarrow \infty} \int_{0}^{1} f\left(g(x)^{n}\right) d x$$

  exists and lies in the interval $[f(0), f(1)]$

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• Paper 4, Section II, F

  #Representation Theory

  2015

  (a) Let $S^{1}$ be the circle group. Assuming any required facts about continuous functions from real analysis, show that every 1-dimensional continuous representation of $S^{1}$ is of the form

  $$z \mapsto z^{n}$$

  for some $n \in \mathbb{Z}$.

  (b) Let $G=S U(2)$, and let $\rho_{V}$ be a continuous representation of $G$ on a finitedimensional vector space $V$.

  (i) Define the character $\chi_{V}$ of $\rho_{V}$, and show that $\chi_{V} \in \mathbb{N}\left[z, z^{-1}\right]$.

  (ii) Show that $\chi_{V}(z)=\chi_{V}\left(z^{-1}\right)$.

  (iii) Let $V$ be the irreducible 4-dimensional representation of $G$. Decompose $V \otimes V$ into irreducible representations. Hence decompose the exterior square $\Lambda^{2} V$ into irreducible representations.

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• Paper 3, Section II, F

  #Representation Theory

  2015

  (a) State Mackey's theorem, defining carefully all the terms used in the statement.

  (b) Let $G$ be a finite group and suppose that $G$ acts on the set $\Omega$.

  If $n \in \mathbb{N}$, we say that the action of $G$ on $\Omega$ is $n$-transitive if $\Omega$ has at least $n$ elements and for every pair of $n$-tuples $\left(a_{1}, \ldots, a_{n}\right)$ and $\left(b_{1}, \ldots, b_{n}\right)$ such that the $a_{i}$ are distinct elements of $\Omega$ and the $b_{i}$ are distinct elements of $\Omega$, there exists $g \in G$ with $g a_{i}=b_{i}$ for every $i$.

  (i) Let $\Omega$ have at least $n$ elements, where $n \geqslant 1$ and let $\omega \in \Omega$. Show that $G$ acts $n$-transitively on $\Omega$ if and only if $G$ acts transitively on $\Omega$ and the stabiliser $G_{\omega}$ acts $(n-1)$-transitively on $\Omega \backslash\{\omega\}$.

  (ii) Show that the permutation module $\mathbb{C} \Omega$ can be decomposed as

  $$\mathbb{C} \Omega=\mathbb{C}_{G} \oplus V,$$

  where $\mathbb{C}_{G}$ is the trivial module and $V$ is some $\mathbb{C} G$-module.

  (iii) Assume that $|\Omega| \geqslant 2$, so that $V \neq 0$. Prove that $V$ is irreducible if and only if $G$ acts 2-transitively on $\Omega$. In that case show also that $V$ is not the trivial representation. [Hint: Pick any orbit of $G$ on $\Omega$; it is isomorphic as a $G$-set to $G / H$ for some subgroup $H \leqslant G$. Consider the induced character $\left.\operatorname{Ind}_{H}^{G} 1_{H} \cdot\right]$

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• Paper 2, Section II, $15 F$

  #Representation Theory

  2015

  Let $G$ be a finite group. Suppose that $\rho: G \rightarrow G \mathrm{G}(V)$ is a finite-dimensional complex representation of dimension $d$. Let $n \in \mathbb{N}$ be arbitrary.

  (i) Define the $n$th symmetric power $S^{n} V$ and the $n$th exterior power $\Lambda^{n} V$ and write down their respective dimensions.

  Let $g \in G$ and let $\lambda_{1}, \ldots, \lambda_{d}$ be the eigenvalues of $g$ on $V$. What are the eigenvalues of $g$ on $S^{n} V$ and on $\Lambda^{n} V$ ?

  (ii) Let $X$ be an indeterminate. For any $g \in G$, define the characteristic polynomial $Q=Q(g, X)$ of $g$ on $V$ by $Q(g, X):=\operatorname{det}(g-X I)$. What is the relationship between the coefficients of $Q$ and the character $\chi_{\Lambda^{n}} V$ of the exterior power?

  Find a relation between the character $\chi_{S^{n}} V$ of the symmetric power and the polynomial $Q$.

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• Paper 1, Section II, F

  #Representation Theory

  2015

  (a) Let $G$ be a finite group and let $\rho: G \rightarrow \mathrm{GL}_{2}(\mathbb{C})$ be a representation of $G$. Suppose that there are elements $g, h$ in $G$ such that the matrices $\rho(g)$ and $\rho(h)$ do not commute. Use Maschke's theorem to prove that $\rho$ is irreducible.

  (b) Let $n$ be a positive integer. You are given that the dicyclic group

  $$G_{4 n}=\left\langle a, b: a^{2 n}=1, a^{n}=b^{2}, b^{-1} a b=a^{-1}\right\rangle$$

  has order $4 n$.

  (i) Show that if $\epsilon$ is any $(2 n)$ th root of unity in $\mathbb{C}$, then there is a representation of $G_{4 n}$ over $\mathbb{C}$ which sends

  $$a \mapsto\left(\begin{array}{cc} \epsilon & 0 \\ 0 & \epsilon^{-1} \end{array}\right), \quad b \mapsto\left(\begin{array}{cc} 0 & 1 \\ \epsilon^{n} & 0 \end{array}\right)$$

  (ii) Find all the irreducible representations of $G_{4 n}$.

  (iii) Find the character table of $G_{4 n}$.

  [Hint: You may find it helpful to consider the cases $n$ odd and $n$ even separately.]

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• Paper 3, Section II, F

  #Riemann Surfaces

  2015

  Let $\wp(z)$ denote the Weierstrass $\wp$-function with respect to a lattice $\Lambda \subset \mathbb{C}$ and let $f$ be an even elliptic function with periods $\Lambda$. Prove that there exists a rational function $Q$ such that $f(z)=Q(\wp(z))$. If we write $Q(w)=p(w) / q(w)$ where $p$ and $q$ are coprime polynomials, find the degree of $f$ in terms of the degrees of the polynomials $p$ and $q$. Describe all even elliptic functions of degree two. Justify your answers. [You may use standard properties of the Weierstrass $\wp$-function.]

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• Paper 2, Section II, F

  #Riemann Surfaces

  2015

  Let $G$ be a domain in $\mathbb{C}$. Define the germ of a function element $(f, D)$ at $z \in D$. Let $\mathcal{G}$ be the set of all germs of function elements in $G$. Define the topology on $\mathcal{G}$. Show it is a topology, and that it is Hausdorff. Define the complex structure on $\mathcal{G}$, and show that there is a natural projection map $\pi: \mathcal{G} \rightarrow G$ which is an analytic covering map on each connected component of $\mathcal{G}$.

  Given a complete analytic function $\mathcal{F}$ on $G$, describe how it determines a connected component $\mathcal{G}_{\mathcal{F}}$ of $\mathcal{G}$. [You may assume that a function element $(g, E)$ is an analytic continuation of a function element $(f, D)$ along a path $\gamma:[0,1] \rightarrow G$ if and only if there is a lift of $\gamma$ to $\mathcal{G}$ starting at the germ of $(f, D)$ at $\gamma(0)$ and ending at the germ of $(g, E)$ at $\gamma(1)$.]

  In each of the following cases, give an example of a domain $G$ in $\mathbb{C}$ and a complete analytic function $\mathcal{F}$ such that:

  (i) $\pi: \mathcal{G}_{\mathcal{F}} \rightarrow G$ is regular but not bijective;

  (ii) $\pi: \mathcal{G}_{\mathcal{F}} \rightarrow G$ is surjective but not regular.

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• Paper 1, Section II, F

  #Riemann Surfaces

  2015

  Let $f: R \rightarrow S$ be a non-constant holomorphic map between compact connected Riemann surfaces and let $B \subset S$ denote the set of branch points. Show that the map $f: R \backslash f^{-1}(B) \rightarrow S \backslash B$ is a regular covering map.

  Given $w \in S \backslash B$ and a closed curve $\gamma$ in $S \backslash B$ with initial and final point $w$, explain how this defines a permutation of the (finite) set $f^{-1}(w)$. Show that the group $H$ obtained from all such closed curves is a transitive subgroup of the full symmetric group of the fibre $f^{-1}(w)$.

  Find the group $H$ for $f: \mathbb{C}_{\infty} \rightarrow \mathbb{C}_{\infty}$ where $f(z)=z^{3} /\left(1-z^{2}\right)$.

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• Paper 4, Section I, J

  #Statistical Modelling

  2015

  Data on 173 nesting female horseshoe crabs record for each crab its colour as one of 4 factors (simply labelled $1, \ldots, 4$ ), its width (in $\mathrm{cm}$ ) and the presence of male crabs nearby (a 1 indicating presence). The data are collected into the $\mathrm{R}$ data frame crabs and the first few lines are displayed below.

  

  Describe the model being fitted by the $\mathrm{R}$ command below.

  $>$ fit1 <- glm(males colour + width, family = binomial, data=crabs)

  The following (abbreviated) output is obtained from the summary command.

  

  Write out the calculation for an approximate $95 \%$ confidence interval for the coefficient for width. Describe the calculation you would perform to obtain an estimate of the probability that a female crab of colour 3 and with a width of $20 \mathrm{~cm}$ has males nearby. [You need not actually compute the end points of the confidence interval or the estimate of the probability above, but merely show the calculations that would need to be performed in order to arrive at them.]

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• Paper 3, Section I, J

  #Statistical Modelling

  2015

  Data are available on the number of counts (atomic disintegration events that take place within a radiation source) recorded with a Geiger counter at a nuclear plant. The counts were registered at each second over a 30 second period for a short-lived, man-made radioactive compound. The first few rows of the dataset are displayed below.

  

  Describe the model being fitted with the following $\mathrm{R}$ command.

  $>$ fit $1<-\operatorname{lm}($ Counts $~$ Time, data=geiger)

  Below is a plot against time of the residuals from the model fitted above.

  

  Referring to the plot, suggest how the model could be improved, and write out the $R$ code for fitting this new model. Briefly describe how one could test in $R$ whether the new model is to be preferred over the old model.

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• Paper 2, Section I, J

  #Statistical Modelling

  2015

  Let $Y_{1}, \ldots, Y_{n}$ be independent Poisson random variables with means $\mu_{1}, \ldots, \mu_{n}$, where $\log \left(\mu_{i}\right)=\beta x_{i}$ for some known constants $x_{i} \in \mathbb{R}$ and an unknown parameter $\beta$. Find the log-likelihood for $\beta$.

  By first computing the first and second derivatives of the log-likelihood for $\beta$, describe the algorithm you would use to find the maximum likelihood estimator $\hat{\beta}$. $[$ Hint: Recall that if $Z \sim \operatorname{Pois}(\mu)$ then

  $$\mathbb{P}(Z=k)=\frac{\mu^{k} e^{-\mu}}{k !}$$

  for $k \in\{0,1,2, \ldots\}$.]

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• Paper 1, Section I, J

  #Statistical Modelling

  2015

  The outputs $Y_{1}, \ldots, Y_{n}$ of a particular process are positive and are believed to be related to $p$-vectors of covariates $x_{1}, \ldots, x_{n}$ according to the following model

  $$\log \left(Y_{i}\right)=\mu+x_{i}^{T} \beta+\varepsilon_{i}$$

  In this model $\varepsilon_{i}$ are i.i.d. $N\left(0, \sigma^{2}\right)$ random variables where $\sigma>0$ is known. It is not possible to measure the output directly, but we can detect whether the output is greater than or less than or equal to a certain known value $c>0$. If

  $$Z_{i}= \begin{cases}1 & \text { if } Y_{i}>c \\ 0 & \text { if } Y_{i} \leqslant c\end{cases}$$

  show that a probit regression model can be used for the data $\left(Z_{i}, x_{i}\right), i=1, \ldots, n$.

  How can we recover $\mu$ and $\beta$ from the parameters of the probit regression model?

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• Paper 4, Section II, J

  #Statistical Modelling

  2015

  Consider the normal linear model where the $n$-vector of responses $Y$ satisfies $Y=X \beta+\varepsilon$ with $\varepsilon \sim N_{n}\left(0, \sigma^{2} I\right)$. Here $X$ is an $n \times p$ matrix of predictors with full column rank where $p \geqslant 3$ and $\beta \in \mathbb{R}^{p}$ is an unknown vector of regression coefficients. For $j \in\{1, \ldots, p\}$, denote the $j$ th column of $X$ by $X_{j}$, and let $X_{-j}$ be $X$ with its $j$ th column removed. Suppose $X_{1}=1_{n}$ where $1_{n}$ is an $n$-vector of 1 's. Denote the maximum likelihood estimate of $\beta$ by $\beta$. Write down the formula for $\hat{\beta}_{j}$ involving $P_{-j}$, the orthogonal projection onto the column space of $X_{-j}$.

  Consider $j, k \in\{2, \ldots, p\}$ with $j<k$. By thinking about the orthogonal projection of $X_{j}$ onto $X_{k}$, show that

  $$\operatorname{var}\left(\hat{\beta}_{j}\right) \geqslant \frac{\sigma^{2}}{\left\|X_{j}\right\|^{2}}\left(1-\left(\frac{X_{k}^{T} X_{j}}{\left\|X_{k}\right\|\left\|X_{j}\right\|}\right)^{2}\right)^{-1}$$

  [You may use standard facts about orthogonal projections including the fact that if $V$ and $W$ are subspaces of $\mathbb{R}^{n}$ with $V$ a subspace of $W$ and $\Pi_{V}$ and $\Pi_{W}$ denote orthogonal projections onto $V$ and $W$ respectively, then for all $v \in \mathbb{R}^{n},\left\|\Pi_{W} v\right\|^{2} \geqslant\left\|\Pi_{V} v\right\|^{2}$.]

  By considering the fitted values $X \hat{\beta}$, explain why if, for any $j \geqslant 2$, a constant is added to each entry in the $j$ th column of $X$, then $\hat{\beta}_{j}$ will remain unchanged. Let $\bar{X}_{j}=\sum_{i=1}^{n} X_{i j} / n$. Why is (*) also true when all instances of $X_{j}$ and $X_{k}$ are replaced by $X_{j}-\bar{X}_{j} 1_{n}$ and $X_{k}-\bar{X}_{k} 1_{n}$ respectively?

  The marks from mid-year statistics and mathematics tests and an end-of-year statistics exam are recorded for 100 secondary school students. The first few lines of the data are given below.

  

  The following abbreviated output is obtained:

  

  What are the hypothesis tests corresponding to the final column of the coefficients table? What is the hypothesis test corresponding to the final line of the output? Interpret the results when testing at the $5 \%$ level.

  How does the following sample correlation matrix for the data help to explain the relative sizes of some of the $p$-values?

  

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• Paper 1, Section II, J

  #Statistical Modelling

  2015

  An experiment is conducted where scientists count the numbers of each of three different strains of fleas that are reproducing in a controlled environment. Varying concentrations of a particular toxin that impairs reproduction are administered to the fleas. The results of the experiment are stored in a data frame $f l e a s$ in $\mathrm{R}$, whose first few rows are given below.

  

  The full dataset has 80 rows. The first column provides the number of fleas, the second provides the concentration of the toxin and the third specifies the strain of the flea as factors 0,1 or 2 . Strain 0 is the common flea and strains 1 and 2 have been genetically modified in a way thought to increase their ability to reproduce in the presence of the toxin.

  Explain and interpret the $\mathrm{R}$ commands and (abbreviated) output below. In particular, you should describe the model being fitted, briefly explain how the standard errors are calculated, and comment on the hypothesis tests being described in the summary.

  

  Explain and motivate the following $\mathrm{R}$ code in the light of the output above. Briefly explain the differences between the models fitted below, and the model corresponding to $f$ it $1 .$

  

  Denote by $M_{1}, M_{2}, M_{3}$ the three models being fitted in sequence above. Explain the hypothesis tests comparing the models to each other that can be performed using the output from the following $R$ code.

  $>c($ fit1$dev, fit2$dev, fit3$dev)

  [1] $56.8756 .9376 .98$

  $>\operatorname{qchisq}(0.95, \mathrm{df}=1)$

  [1] $3.84$

  Use these numbers to comment on the most appropriate model for the data.

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• Paper 4, Section II, C

  #Statistical Physics

  2015

  The Ising model consists of $N$ particles, labelled by $i$, arranged on a $D$-dimensional Euclidean lattice with periodic boundary conditions. Each particle has spin up $s_{i}=+1$, or down $s_{i}=-1$, and the energy in the presence of a magnetic field $B$ is

  $$E=-B \sum_{i} s_{i}-J \sum_{\langle i, j\rangle} s_{i} s_{j}$$

  where $J>0$ is a constant and $\langle i, j\rangle$ indicates that the second sum is over each pair of nearest neighbours (every particle has $2 D$ nearest neighbours). Let $\beta=1 / k_{B} T$, where $T$ is the temperature.

  (i) Express the average spin per particle, $m=\left(\sum_{i}\left\langle s_{i}\right\rangle\right) / N$, in terms of the canonical partition function $Z$.

  (ii) Show that in the mean-field approximation

  $$Z=C\left[Z_{1}\left(\beta B_{\mathrm{eff}}\right)\right]^{N}$$

  where $Z_{1}$ is a single-particle partition function, $B_{\text {eff }}$ is an effective magnetic field which you should find in terms of $B, J, D$ and $m$, and $C$ is a prefactor which you should also evaluate.

  (iii) Deduce an equation that determines $m$ for general values of $B, J$ and temperature $T$. Without attempting to solve for $m$ explicitly, discuss how the behaviour of the system depends on temperature when $B=0$, deriving an expression for the critical temperature $T_{c}$ and explaining its significance.

  (iv) Comment briefly on whether the results obtained using the mean-field approximation for $B=0$ are consistent with an expression for the free energy of the form

  $$F(m, T)=F_{0}(T)+\frac{a}{2}\left(T-T_{c}\right) m^{2}+\frac{b}{4} m^{4}$$

  where $a$ and $b$ are positive constants.

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• Paper 3, Section II, C

  #Statistical Physics

  2015

  (a) A sample of gas has pressure $p$, volume $V$, temperature $T$ and entropy $S$.

  (i) Use the first law of thermodynamics to derive the Maxwell relation

  $$\left(\frac{\partial S}{\partial p}\right)_{T}=-\left(\frac{\partial V}{\partial T}\right)_{p} .$$

  (ii) Define the heat capacity at constant pressure $C_{p}$ and the enthalpy $H$ and show that $C_{p}=(\partial H / \partial T)_{p}$.

  (b) Consider a perfectly insulated pipe with a throttle valve, as shown.

  

  Gas initially occupying volume $V_{1}$ on the left is forced slowly through the valve at constant pressure $p_{1}$. A constant pressure $p_{2}$ is maintained on the right and the final volume occupied by the gas after passing through the valve is $V_{2}$.

  (i) Show that the enthalpy $H$ of the gas is unchanged by this process.

  (ii) The Joule-Thomson coefficient is defined to be $\mu=(\partial T / \partial p)_{H}$. Show that

  $$\mu=\frac{V}{C_{p}}\left[\frac{T}{V}\left(\frac{\partial V}{\partial T}\right)_{p}-1\right]$$

  [You may assume the identity $(\partial y / \partial x)_{u}=-(\partial u / \partial x)_{y} /(\partial u / \partial y)_{x} \cdot$ ]

  (iii) Suppose that the gas obeys an equation of state

  $$p=k_{B} T\left[\frac{N}{V}+B_{2}(T) \frac{N^{2}}{V^{2}}\right]$$

  where $N$ is the number of particles. Calculate $\mu$ to first order in $N / V$ and hence derive a condition on $\frac{d}{d T}\left(\frac{B_{2}(T)}{T}\right)$ for obtaining a positive Joule-Thomson coefficient.

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• Paper 2, Section II, C

  #Statistical Physics

  2015

  (a) State the Bose-Einstein distribution formula for the mean occupation numbers $n_{i}$ of discrete single-particle states $i$ with energies $E_{i}$ in a gas of bosons. Write down expressions for the total particle number $N$ and the total energy $U$ when the singleparticle states can be treated as continuous, with energies $E \geqslant 0$ and density of states $g(E)$.

  (b) Blackbody radiation at temperature $T$ is equivalent to a gas of photons with

  $$g(E)=A V E^{2}$$

  where $V$ is the volume and $A$ is a constant. What value of the chemical potential is required when applying the Bose-Einstein distribution to photons? Show that the heat capacity at constant volume satisfies $C_{V} \propto T^{\alpha}$ for some constant $\alpha$, to be determined.

  (c) Consider a system of bosonic particles of fixed total number $N \gg 1$. The particles are trapped in a potential which has ground state energy zero and which gives rise to a density of states $g(E)=B E^{2}$, where $B$ is a constant. Explain, for this system, what is meant by Bose-Einstein condensation and show that the critical temperature satisfies $T_{c} \propto N^{1 / 3}$. If $N_{0}$ is the number of particles in the ground state, show that for $T$ just below $T_{c}$

  $$N_{0} / N \approx 1-\left(T / T_{c}\right)^{\gamma}$$

  for some constant $\gamma$, to be determined.

  (d) Would you expect photons to exhibit Bose-Einstein condensation? Explain your answer very briefly.

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• Paper 1, Section II, C

  #Statistical Physics

  2015

  (a) Define the canonical partition function $Z$ for a system with energy levels $E_{n}$, where $n$ labels states, given that the system is in contact with a heat reservoir at temperature $T$. What is the probability $p(n)$ that the system occupies state $n$ ? Starting from an expression for the entropy $S=k_{B} \partial(T \ln Z) / \partial T$, deduce that

  $$S=-k_{B} \sum_{n} p(n) \ln p(n)$$

  (b) Consider an ensemble consisting of $W$ copies of the system in part (a) with $W$ very large, so that there are $W p(n)$ members of the ensemble in state $n$. Starting from an expression for the number of ways in which this can occur, find the entropy $S_{W}$ of the ensemble and hence re-derive the expression $(*)$. [You may assume Stirling's formula $\ln X ! \approx X \ln X-X$ for $X$ large. $]$

  (c) Consider a system of $N$ non-interacting particles at temperature $T$. Each particle has $q$ internal states with energies

  $$0, \mathcal{E}, 2 \mathcal{E}, \ldots,(q-1) \mathcal{E}$$

  Assuming that the internal states are the only relevant degrees of freedom, calculate the total entropy of the system. Find the limiting values of the entropy as $T \rightarrow 0$ and $T \rightarrow \infty$ and comment briefly on your answers.

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• Paper 4, Section II, $26 \mathrm{~K}$

  #Stochastic Financial Models

  2015

  (i) An investor in a single-period market with time- 0 wealth $w_{0}$ may generate any time-1 wealth $w_{1}$ of the form $w_{1}=w_{0}+X$, where $X$ is any element of a vector space $V$ of random variables. The investor's objective is to maximize $E\left[U\left(w_{1}\right)\right]$, where $U$ is strictly increasing, concave and $C^{2}$. Define the utility indifference price $\pi(Y)$ of a random variable $Y$.

  Prove that the map $Y \mapsto \pi(Y)$ is concave. [You may assume that any supremum is attained.]

  (ii) Agent $j$ has utility $U_{j}(x)=-\exp \left(-\gamma_{j} x\right), j=1, \ldots, J$. The agents may buy for time- 0 price $p$ a risky asset which will be worth $X$ at time 1 , where $X$ is random and has density

  $$f(x)=\frac{1}{2} \alpha e^{-\alpha|x|}, \quad-\infty<x<\infty .$$

  Assuming zero interest, prove that agent $j$ will optimally choose to buy

  $$\theta_{j}=-\frac{\sqrt{1+p^{2} \alpha^{2}}-1}{\gamma_{j} p}$$

  units of the risky asset at time 0 .

  If the asset is in unit net supply, if $\Gamma^{-1} \equiv \sum_{j} \gamma_{j}^{-1}$, and if $\alpha>\Gamma$, prove that the market for the risky asset will clear at price

  $$p=-\frac{2 \Gamma}{\alpha^{2}-\Gamma^{2}}$$

  What happens if $\alpha \leqslant \Gamma ?$

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• Paper 3, Section II, $26 \mathrm{~K}$

  #Stochastic Financial Models

  2015

  A single-period market consists of $n$ assets whose prices at time $t$ are denoted by $S_{t}=\left(S_{t}^{1}, \ldots, S_{t}^{n}\right)^{T}, t=0,1$, and a riskless bank account bearing interest rate $r$. The value of $S_{0}$ is given, and $S_{1} \sim N(\mu, V)$. An investor with utility $U(x)=-\exp (-\gamma x)$ wishes to choose a portfolio of the available assets so as to maximize the expected utility of her wealth at time 1. Find her optimal investment.

  What is the market portfolio for this problem? What is the beta of asset $i$ ? Derive the Capital Asset Pricing Model, that

  Excess return of asset $i=$ Excess return of market portfolio $\times \beta_{i}$.

  The Sharpe ratio of a portfolio $\theta$ is defined to be the excess return of the portfolio $\theta$ divided by the standard deviation of the portfolio $\theta$. If $\rho_{i}$ is the correlation of the return on asset $i$ with the return on the market portfolio, prove that

  Sharpe ratio of asset $i=$ Sharpe ratio of market portfolio $\times \rho_{i}$.

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• Paper 1, Section II, $26 K$

  #Stochastic Financial Models

  2015

  (i) What does it mean to say that $\left(X_{n}, \mathcal{F}_{n}\right)_{n \geqslant 0}$ is a martingale?

  (ii) If $Y$ is an integrable random variable and $Y_{n}=E\left[Y \mid \mathcal{F}_{n}\right]$, prove that $\left(Y_{n}, \mathcal{F}_{n}\right)$ is a martingale. [Standard facts about conditional expectation may be used without proof provided they are clearly stated.] When is it the case that the limit $\lim _{n \rightarrow \infty} Y_{n}$ exists almost surely?

  (iii) An urn contains initially one red ball and one blue ball. A ball is drawn at random and then returned to the urn with a new ball of the other colour. This process is repeated, adding one ball at each stage to the urn. If the number of red balls after $n$ draws and replacements is $X_{n}$, and the number of blue balls is $Y_{n}$, show that $M_{n}=h\left(X_{n}, Y_{n}\right)$ is a martingale, where

  $$h(x, y)=(x-y)(x+y-1)$$

  Does this martingale converge almost surely?

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• Paper 2, Section II, $27 \mathrm{~K}$

  #Stochastic Financial Models

  2015

  (i) What is Brownian motion?

  (ii) Suppose that $\left(B_{t}\right)_{t \geqslant 0}$ is Brownian motion, and the price $S_{t}$ at time $t$ of a risky asset is given by

  $$S_{t}=S_{0} \exp \left\{\sigma B_{t}+\left(\mu-\frac{1}{2} \sigma^{2}\right) t\right\}$$

  where $\mu>0$ is the constant growth rate, and $\sigma>0$ is the constant volatility of the asset. Assuming that the riskless rate of interest is $r>0$, derive an expression for the price at time 0 of a European call option with strike $K$ and expiry $T$, explaining briefly the basis for your calculation.

  (iii) With the same notation, derive the time-0 price of a European option with expiry $T$ which at expiry pays

  $$\left\{\left(S_{T}-K\right)^{+}\right\}^{2} / S_{T}$$

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• Paper 4, Section I, $2 I$

  #Topics in Analysis

  2015

  Let $\mathcal{K}$ be the set of all non-empty compact subsets of $m$-dimensional Euclidean space $\mathbb{R}^{m}$. Define the Hausdorff metric on $\mathcal{K}$, and prove that it is a metric.

  Let $K_{1} \supseteq K_{2} \supseteq \ldots$ be a sequence in $\mathcal{K}$. Show that $K=\bigcap_{n=1}^{\infty} K_{n}$ is also in $\mathcal{K}$ and that $K_{n} \rightarrow K$ as $n \rightarrow \infty$ in the Hausdorff metric.

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• Paper 3, Section I, $2 I$

  #Topics in Analysis

  2015

  Let $K$ be a compact subset of $\mathbb{C}$ with path-connected complement. If $w \notin K$ and $\epsilon>0$, show that there is a polynomial $P$ such that

  $$\left|P(z)-\frac{1}{w-z}\right| \leqslant \epsilon$$

  for all $z \in K$.

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• Paper 2, Section I, I

  #Topics in Analysis

  2015

  Let $x_{1}, x_{2}, \ldots, x_{n}$ be the roots of the Legendre polynomial of degree $n$. Let $A_{1}$, $A_{2}, \ldots, A_{n}$ be chosen so that

  $$\int_{-1}^{1} p(t) d t=\sum_{j=1}^{n} A_{j} p\left(x_{j}\right)$$

  for all polynomials $p$ of degree $n-1$ or less. Assuming any results about Legendre polynomials that you need, prove the following results:

  (i) $\int_{-1}^{1} p(t) d t=\sum_{j=1}^{n} A_{j} p\left(x_{j}\right)$ for all polynomials $p$ of degree $2 n-1$ or less;

  (ii) $A_{j} \geqslant 0$ for all $1 \leqslant j \leqslant n$;

  (iii) $\sum_{j=1}^{n} A_{j}=2$.

  Now consider $Q_{n}(f)=\sum_{j=1}^{n} A_{j} f\left(x_{j}\right)$. Show that

  $$Q_{n}(f) \rightarrow \int_{-1}^{1} f(t) d t$$

  as $n \rightarrow \infty$ for all continuous functions $f$.

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• Paper 1, Section I, I

  #Topics in Analysis

  2015

  Let $\Omega$ be a non-empty bounded open subset of $\mathbb{R}^{2}$ with closure $\bar{\Omega}$ and boundary $\partial \Omega$. Let $\phi: \bar{\Omega} \rightarrow \mathbb{R}$ be continuous with $\phi$ twice differentiable on $\Omega$.

  (i) Why does $\phi$ have a maximum on $\bar{\Omega}$ ?

  (ii) If $\epsilon>0$ and $\nabla^{2} \phi \geqslant \epsilon$ on $\Omega$, show that $\phi$ has a maximum on $\partial \Omega$.

  (iii) If $\nabla^{2} \phi \geqslant 0$ on $\Omega$, show that $\phi$ has a maximum on $\partial \Omega$.

  (iv) If $\nabla^{2} \phi=0$ on $\Omega$ and $\phi=0$ on $\partial \Omega$, show that $\phi=0$ on $\bar{\Omega}$.

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• Paper 2, Section II,

  #Topics in Analysis

  2015

  State and prove Sperner's lemma concerning the colouring of triangles.

  Deduce a theorem, to be stated clearly, on retractions to the boundary of a disc.

  State Brouwer's fixed point theorem for a disc and sketch a proof of it.

  Let $g: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ be a continuous function such that for some $K>0$ we have $\|g(x)-x\| \leqslant K$ for all $x \in \mathbb{R}^{2}$. Show that $g$ is surjective.

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• Paper 3, Section II, I

  #Topics in Analysis

  2015

  Let $\alpha>0$. By considering the set $E_{m}$ consisting of those $f \in C([0,1])$ for which there exists an $x \in[0,1]$ with $|f(x+h)-f(x)| \leqslant m|h|^{\alpha}$ for all $x+h \in[0,1]$, or otherwise, give a Baire category proof of the existence of continuous functions $f$ on $[0,1]$ such that

  $$\limsup _{h \rightarrow 0}|h|^{-\alpha}|f(x+h)-f(x)|=\infty$$

  at each $x \in[0,1]$.

  Are the following statements true? Give reasons.

  (i) There exists an $f \in C([0,1])$ such that

  $$\limsup _{h \rightarrow 0}|h|^{-\alpha}|f(x+h)-f(x)|=\infty$$

  for each $x \in[0,1]$ and each $\alpha>0$.

  (ii) There exists an $f \in C([0,1])$ such that

  $$\limsup _{h \rightarrow 0}|h|^{-\alpha}|f(x+h)-f(x)|=\infty$$

  for each $x \in[0,1]$ and each $\alpha \geqslant 0$.

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• Paper 4, Section II, 36B

  #Waves

  2015

  The shallow-water equations

  $$\frac{\partial h}{\partial t}+u \frac{\partial h}{\partial x}+h \frac{\partial u}{\partial x}=0, \quad \frac{\partial u}{\partial t}+u \frac{\partial u}{\partial x}+g \frac{\partial h}{\partial x}=0$$

  describe one-dimensional flow over a horizontal boundary with depth $h(x, t)$ and velocity $u(x, t)$, where $g$ is the acceleration due to gravity.

  Show that the Riemann invariants $u \pm 2\left(c-c_{0}\right)$ are constant along characteristics $C_{\pm}$satisfying $d x / d t=u \pm c$, where $c(h)$ is the linear wave speed and $c_{0}$ denotes a reference state.

  An initially stationary pool of fluid of depth $h_{0}$ is held between a stationary wall at $x=a>0$ and a removable barrier at $x=0$. At $t=0$ the barrier is instantaneously removed allowing the fluid to flow into the region $x<0$.

  For $0 \leqslant t \leqslant a / c_{0}$, find $u(x, t)$ and $c(x, t)$ in each of the regions

  $$\begin{gathered} \text { (i) } \quad c_{0} t \leqslant x \leqslant a \\ \text { (ii) }-2 c_{0} t \leqslant x \leqslant c_{0} t \end{gathered}$$

  explaining your argument carefully with a sketch of the characteristics in the $(x, t)$ plane.

  For $t \geqslant a / c_{0}$, show that the solution in region (ii) above continues to hold in the region $-2 c_{0} t \leqslant x \leqslant 3 a\left(c_{0} t / a\right)^{1 / 3}-2 c_{0} t$. Explain why this solution does not hold in $3 a\left(c_{0} t / a\right)^{1 / 3}-2 c_{0} t<x<a .$

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• Paper 2, Section II, 36B

  #Waves

  2015

  A uniform elastic solid with density $\rho$ and Lamé moduli $\lambda$ and $\mu$ occupies the region between rigid plane boundaries $z=0$ and $z=h$. Starting with the linear elastic wave equation, show that SH waves can propagate in the $x$-direction within this waveguide, and find the dispersion relation $\omega(k)$ for the various modes.

  State the cut-off frequency for each mode. Find the corresponding phase velocity $c(k)$ and group velocity $c_{g}(k)$, and sketch these functions for $k, \omega>0$.

  Define the time and cross-sectional average appropriate for a mode with frequency $\mathrm{~}$ energy. [You may assume that the elastic energy per unit volume is $\frac{1}{2}\left(\lambda e_{k k}^{2}+2 \mu e_{i j} e_{i j}\right)$.]

  An elastic displacement of the form $\mathbf{u}=(0, f(x, z), 0)$ is created in a region near $x=0$, and then released at $t=0$. Explain briefly how the amplitude of the resulting disturbance varies with time as $t \rightarrow \infty$ at the moving position $x=V t$ for each of the cases $0<V^{2}<\mu / \rho$ and $V^{2}>\mu / \rho$. [You may quote without proof any generic results from the method of stationary phase.]

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• Paper 3, Section II, B

  #Waves

  2015

  Derive the ray-tracing equations for the quantities $\mathrm{d} k_{i} / \mathrm{d} t, \mathrm{~d} \omega / \mathrm{d} t$ and $\mathrm{d} x_{i} / \mathrm{d} t$ during wave propagation through a slowly varying medium with local dispersion relation $\omega=\Omega(\mathbf{k}, \mathbf{x}, t)$, explaining the meaning of the notation $\mathrm{d} / \mathrm{d} t$.

  The dispersion relation for water waves is $\Omega^{2}=g \kappa \tanh (\kappa h)$, where $h$ is the water depth, $\kappa^{2}=k^{2}+l^{2}$, and $k$ and $l$ are the components of $\mathbf{k}$ in the horizontal $x$ and $y$ directions. Water waves are incident from an ocean occupying $x>0,-\infty<y<\infty$ onto a beach at $x=0$. The undisturbed water depth is $h(x)=\alpha x^{p}$, where $\alpha, p$ are positive constants and $\alpha$ is sufficiently small that the depth can be assumed to be slowly varying. Far from the beach, the waves are planar with frequency $\omega_{\infty}$ and with crests making an acute angle $\theta_{\infty}$ with the shoreline.

  Obtain a differential equation (with $k$ defined implicitly) for a ray $y=y(x)$ and show that near the shore the ray satisfies

  $$y-y_{0} \sim A x^{q}$$

  where $A$ and $q$ should be found. Sketch the shape of the wavecrests near the shoreline for the case $p<2$.

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• Paper 1, Section II, 37B

  #Waves

  2015

  An acoustic plane wave (not necessarily harmonic) travels at speed $c_{0}$ in the direction $\hat{\mathbf{k}}$, where $|\hat{\mathbf{k}}|=1$, through an inviscid, compressible fluid of unperturbed density $\rho_{0}$. Show that the velocity $\tilde{\mathbf{u}}$ is proportional to the perturbation pressure $\tilde{p}$, and find $\tilde{\mathbf{u}} / \tilde{p}$. Define the acoustic intensity $\mathbf{I}$.

  A harmonic acoustic plane wave with wavevector $\mathbf{k}=k(\cos \theta, \sin \theta, 0)$ and unitamplitude perturbation pressure is incident from $x<0$ on a thin elastic membrane at unperturbed position $x=0$. The regions $x<0$ and $x>0$ are both occupied by gas with density $\rho_{0}$ and sound speed $c_{0}$. The kinematic boundary conditions at the membrane are those appropriate for an inviscid fluid, and the (linearized) dynamic boundary condition

  $$m \frac{\partial^{2} X}{\partial t^{2}}-T \frac{\partial^{2} X}{\partial y^{2}}+[\tilde{p}(0, y, t)]_{-}^{+}=0$$

  where $T$ and $m$ are the tension and mass per unit area of the membrane, and $x=X(y, t)$ (with $|k X| \ll 1$ ) is its perturbed position. Find the amplitudes of the reflected and transmitted pressure perturbations, expressing your answers in terms of the dimensionless parameter

  $$\alpha=\frac{\rho_{0} c_{0}^{2}}{k \cos \theta\left(m c_{0}^{2}-T \sin ^{2} \theta\right)} .$$

  Hence show that the time-averaged energy flux in the $x$-direction is conserved across the membrane.

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• Paper 2, Section II, A

  #Variational Principles

  2015

  A right circular cylinder of radius $a$ and length $l$ has volume $V$ and total surface area $A$. Use Lagrange multipliers to do the following:

  (a) Show that, for a given total surface area, the maximum volume is

  $$V=\frac{1}{3} \sqrt{\frac{A^{3}}{C \pi}}$$

  determining the integer $C$ in the process.

  (b) For a cylinder inscribed in the unit sphere, show that the value of $l / a$ which maximises the area of the cylinder is

  $$D+\sqrt{E}$$

  determining the integers $D$ and $E$ as you do so.

  (c) Consider the rectangular parallelepiped of largest volume which fits inside a hemisphere of fixed radius. Find the ratio of the parallelepiped's volume to the volume of the hemisphere.

  [You need not show that suitable extrema you find are actually maxima.]

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• Paper 4, Section II, A

  #Variational Principles

  2015

  Derive the Euler-Lagrange equation for the integral

  $$\int_{x_{0}}^{x_{1}} f\left(x, u, u^{\prime}\right) d x$$

  where $u\left(x_{0}\right)$ is allowed to float, $\partial f /\left.\partial u^{\prime}\right|_{x_{0}}=0$ and $u\left(x_{1}\right)$ takes a given value.

  Given that $y(0)$ is finite, $y(1)=1$ and $y^{\prime}(1)=1$, find the stationary value of

  $$J=\int_{0}^{1}\left(x^{4}\left(y^{\prime \prime}\right)^{2}+4 x^{2}\left(y^{\prime}\right)^{2}\right) d x$$

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